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Diffstat (limited to 'eigen/Eigen/src/OrderingMethods/Eigen_Colamd.h')
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diff --git a/eigen/Eigen/src/OrderingMethods/Eigen_Colamd.h b/eigen/Eigen/src/OrderingMethods/Eigen_Colamd.h new file mode 100644 index 0000000..359fd44 --- /dev/null +++ b/eigen/Eigen/src/OrderingMethods/Eigen_Colamd.h @@ -0,0 +1,1843 @@ +// // This file is part of Eigen, a lightweight C++ template library +// for linear algebra. +// +// Copyright (C) 2012 Desire Nuentsa Wakam <desire.nuentsa_wakam@inria.fr> +// +// This Source Code Form is subject to the terms of the Mozilla +// Public License v. 2.0. If a copy of the MPL was not distributed +// with this file, You can obtain one at http://mozilla.org/MPL/2.0/. + +// This file is modified from the colamd/symamd library. The copyright is below + +// The authors of the code itself are Stefan I. Larimore and Timothy A. +// Davis (davis@cise.ufl.edu), University of Florida. The algorithm was +// developed in collaboration with John Gilbert, Xerox PARC, and Esmond +// Ng, Oak Ridge National Laboratory. +// +// Date: +// +// September 8, 2003. Version 2.3. +// +// Acknowledgements: +// +// This work was supported by the National Science Foundation, under +// grants DMS-9504974 and DMS-9803599. +// +// Notice: +// +// Copyright (c) 1998-2003 by the University of Florida. +// All Rights Reserved. +// +// THIS MATERIAL IS PROVIDED AS IS, WITH ABSOLUTELY NO WARRANTY +// EXPRESSED OR IMPLIED. ANY USE IS AT YOUR OWN RISK. +// +// Permission is hereby granted to use, copy, modify, and/or distribute +// this program, provided that the Copyright, this License, and the +// Availability of the original version is retained on all copies and made +// accessible to the end-user of any code or package that includes COLAMD +// or any modified version of COLAMD. +// +// Availability: +// +// The colamd/symamd library is available at +// +// http://www.suitesparse.com + + +#ifndef EIGEN_COLAMD_H +#define EIGEN_COLAMD_H + +namespace internal { +/* Ensure that debugging is turned off: */ +#ifndef COLAMD_NDEBUG +#define COLAMD_NDEBUG +#endif /* NDEBUG */ +/* ========================================================================== */ +/* === Knob and statistics definitions ====================================== */ +/* ========================================================================== */ + +/* size of the knobs [ ] array. Only knobs [0..1] are currently used. */ +#define COLAMD_KNOBS 20 + +/* number of output statistics. Only stats [0..6] are currently used. */ +#define COLAMD_STATS 20 + +/* knobs [0] and stats [0]: dense row knob and output statistic. */ +#define COLAMD_DENSE_ROW 0 + +/* knobs [1] and stats [1]: dense column knob and output statistic. */ +#define COLAMD_DENSE_COL 1 + +/* stats [2]: memory defragmentation count output statistic */ +#define COLAMD_DEFRAG_COUNT 2 + +/* stats [3]: colamd status: zero OK, > 0 warning or notice, < 0 error */ +#define COLAMD_STATUS 3 + +/* stats [4..6]: error info, or info on jumbled columns */ +#define COLAMD_INFO1 4 +#define COLAMD_INFO2 5 +#define COLAMD_INFO3 6 + +/* error codes returned in stats [3]: */ +#define COLAMD_OK (0) +#define COLAMD_OK_BUT_JUMBLED (1) +#define COLAMD_ERROR_A_not_present (-1) +#define COLAMD_ERROR_p_not_present (-2) +#define COLAMD_ERROR_nrow_negative (-3) +#define COLAMD_ERROR_ncol_negative (-4) +#define COLAMD_ERROR_nnz_negative (-5) +#define COLAMD_ERROR_p0_nonzero (-6) +#define COLAMD_ERROR_A_too_small (-7) +#define COLAMD_ERROR_col_length_negative (-8) +#define COLAMD_ERROR_row_index_out_of_bounds (-9) +#define COLAMD_ERROR_out_of_memory (-10) +#define COLAMD_ERROR_internal_error (-999) + +/* ========================================================================== */ +/* === Definitions ========================================================== */ +/* ========================================================================== */ + +#define ONES_COMPLEMENT(r) (-(r)-1) + +/* -------------------------------------------------------------------------- */ + +#define COLAMD_EMPTY (-1) + +/* Row and column status */ +#define ALIVE (0) +#define DEAD (-1) + +/* Column status */ +#define DEAD_PRINCIPAL (-1) +#define DEAD_NON_PRINCIPAL (-2) + +/* Macros for row and column status update and checking. */ +#define ROW_IS_DEAD(r) ROW_IS_MARKED_DEAD (Row[r].shared2.mark) +#define ROW_IS_MARKED_DEAD(row_mark) (row_mark < ALIVE) +#define ROW_IS_ALIVE(r) (Row [r].shared2.mark >= ALIVE) +#define COL_IS_DEAD(c) (Col [c].start < ALIVE) +#define COL_IS_ALIVE(c) (Col [c].start >= ALIVE) +#define COL_IS_DEAD_PRINCIPAL(c) (Col [c].start == DEAD_PRINCIPAL) +#define KILL_ROW(r) { Row [r].shared2.mark = DEAD ; } +#define KILL_PRINCIPAL_COL(c) { Col [c].start = DEAD_PRINCIPAL ; } +#define KILL_NON_PRINCIPAL_COL(c) { Col [c].start = DEAD_NON_PRINCIPAL ; } + +/* ========================================================================== */ +/* === Colamd reporting mechanism =========================================== */ +/* ========================================================================== */ + +// == Row and Column structures == +template <typename Index> +struct colamd_col +{ + Index start ; /* index for A of first row in this column, or DEAD */ + /* if column is dead */ + Index length ; /* number of rows in this column */ + union + { + Index thickness ; /* number of original columns represented by this */ + /* col, if the column is alive */ + Index parent ; /* parent in parent tree super-column structure, if */ + /* the column is dead */ + } shared1 ; + union + { + Index score ; /* the score used to maintain heap, if col is alive */ + Index order ; /* pivot ordering of this column, if col is dead */ + } shared2 ; + union + { + Index headhash ; /* head of a hash bucket, if col is at the head of */ + /* a degree list */ + Index hash ; /* hash value, if col is not in a degree list */ + Index prev ; /* previous column in degree list, if col is in a */ + /* degree list (but not at the head of a degree list) */ + } shared3 ; + union + { + Index degree_next ; /* next column, if col is in a degree list */ + Index hash_next ; /* next column, if col is in a hash list */ + } shared4 ; + +}; + +template <typename Index> +struct Colamd_Row +{ + Index start ; /* index for A of first col in this row */ + Index length ; /* number of principal columns in this row */ + union + { + Index degree ; /* number of principal & non-principal columns in row */ + Index p ; /* used as a row pointer in init_rows_cols () */ + } shared1 ; + union + { + Index mark ; /* for computing set differences and marking dead rows*/ + Index first_column ;/* first column in row (used in garbage collection) */ + } shared2 ; + +}; + +/* ========================================================================== */ +/* === Colamd recommended memory size ======================================= */ +/* ========================================================================== */ + +/* + The recommended length Alen of the array A passed to colamd is given by + the COLAMD_RECOMMENDED (nnz, n_row, n_col) macro. It returns -1 if any + argument is negative. 2*nnz space is required for the row and column + indices of the matrix. colamd_c (n_col) + colamd_r (n_row) space is + required for the Col and Row arrays, respectively, which are internal to + colamd. An additional n_col space is the minimal amount of "elbow room", + and nnz/5 more space is recommended for run time efficiency. + + This macro is not needed when using symamd. + + Explicit typecast to Index added Sept. 23, 2002, COLAMD version 2.2, to avoid + gcc -pedantic warning messages. +*/ +template <typename Index> +inline Index colamd_c(Index n_col) +{ return Index( ((n_col) + 1) * sizeof (colamd_col<Index>) / sizeof (Index) ) ; } + +template <typename Index> +inline Index colamd_r(Index n_row) +{ return Index(((n_row) + 1) * sizeof (Colamd_Row<Index>) / sizeof (Index)); } + +// Prototypes of non-user callable routines +template <typename Index> +static Index init_rows_cols (Index n_row, Index n_col, Colamd_Row<Index> Row [], colamd_col<Index> col [], Index A [], Index p [], Index stats[COLAMD_STATS] ); + +template <typename Index> +static void init_scoring (Index n_row, Index n_col, Colamd_Row<Index> Row [], colamd_col<Index> Col [], Index A [], Index head [], double knobs[COLAMD_KNOBS], Index *p_n_row2, Index *p_n_col2, Index *p_max_deg); + +template <typename Index> +static Index find_ordering (Index n_row, Index n_col, Index Alen, Colamd_Row<Index> Row [], colamd_col<Index> Col [], Index A [], Index head [], Index n_col2, Index max_deg, Index pfree); + +template <typename Index> +static void order_children (Index n_col, colamd_col<Index> Col [], Index p []); + +template <typename Index> +static void detect_super_cols (colamd_col<Index> Col [], Index A [], Index head [], Index row_start, Index row_length ) ; + +template <typename Index> +static Index garbage_collection (Index n_row, Index n_col, Colamd_Row<Index> Row [], colamd_col<Index> Col [], Index A [], Index *pfree) ; + +template <typename Index> +static inline Index clear_mark (Index n_row, Colamd_Row<Index> Row [] ) ; + +/* === No debugging ========================================================= */ + +#define COLAMD_DEBUG0(params) ; +#define COLAMD_DEBUG1(params) ; +#define COLAMD_DEBUG2(params) ; +#define COLAMD_DEBUG3(params) ; +#define COLAMD_DEBUG4(params) ; + +#define COLAMD_ASSERT(expression) ((void) 0) + + +/** + * \brief Returns the recommended value of Alen + * + * Returns recommended value of Alen for use by colamd. + * Returns -1 if any input argument is negative. + * The use of this routine or macro is optional. + * Note that the macro uses its arguments more than once, + * so be careful for side effects, if you pass expressions as arguments to COLAMD_RECOMMENDED. + * + * \param nnz nonzeros in A + * \param n_row number of rows in A + * \param n_col number of columns in A + * \return recommended value of Alen for use by colamd + */ +template <typename Index> +inline Index colamd_recommended ( Index nnz, Index n_row, Index n_col) +{ + if ((nnz) < 0 || (n_row) < 0 || (n_col) < 0) + return (-1); + else + return (2 * (nnz) + colamd_c (n_col) + colamd_r (n_row) + (n_col) + ((nnz) / 5)); +} + +/** + * \brief set default parameters The use of this routine is optional. + * + * Colamd: rows with more than (knobs [COLAMD_DENSE_ROW] * n_col) + * entries are removed prior to ordering. Columns with more than + * (knobs [COLAMD_DENSE_COL] * n_row) entries are removed prior to + * ordering, and placed last in the output column ordering. + * + * COLAMD_DENSE_ROW and COLAMD_DENSE_COL are defined as 0 and 1, + * respectively, in colamd.h. Default values of these two knobs + * are both 0.5. Currently, only knobs [0] and knobs [1] are + * used, but future versions may use more knobs. If so, they will + * be properly set to their defaults by the future version of + * colamd_set_defaults, so that the code that calls colamd will + * not need to change, assuming that you either use + * colamd_set_defaults, or pass a (double *) NULL pointer as the + * knobs array to colamd or symamd. + * + * \param knobs parameter settings for colamd + */ + +static inline void colamd_set_defaults(double knobs[COLAMD_KNOBS]) +{ + /* === Local variables ================================================== */ + + int i ; + + if (!knobs) + { + return ; /* no knobs to initialize */ + } + for (i = 0 ; i < COLAMD_KNOBS ; i++) + { + knobs [i] = 0 ; + } + knobs [COLAMD_DENSE_ROW] = 0.5 ; /* ignore rows over 50% dense */ + knobs [COLAMD_DENSE_COL] = 0.5 ; /* ignore columns over 50% dense */ +} + +/** + * \brief Computes a column ordering using the column approximate minimum degree ordering + * + * Computes a column ordering (Q) of A such that P(AQ)=LU or + * (AQ)'AQ=LL' have less fill-in and require fewer floating point + * operations than factorizing the unpermuted matrix A or A'A, + * respectively. + * + * + * \param n_row number of rows in A + * \param n_col number of columns in A + * \param Alen, size of the array A + * \param A row indices of the matrix, of size ALen + * \param p column pointers of A, of size n_col+1 + * \param knobs parameter settings for colamd + * \param stats colamd output statistics and error codes + */ +template <typename Index> +static bool colamd(Index n_row, Index n_col, Index Alen, Index *A, Index *p, double knobs[COLAMD_KNOBS], Index stats[COLAMD_STATS]) +{ + /* === Local variables ================================================== */ + + Index i ; /* loop index */ + Index nnz ; /* nonzeros in A */ + Index Row_size ; /* size of Row [], in integers */ + Index Col_size ; /* size of Col [], in integers */ + Index need ; /* minimum required length of A */ + Colamd_Row<Index> *Row ; /* pointer into A of Row [0..n_row] array */ + colamd_col<Index> *Col ; /* pointer into A of Col [0..n_col] array */ + Index n_col2 ; /* number of non-dense, non-empty columns */ + Index n_row2 ; /* number of non-dense, non-empty rows */ + Index ngarbage ; /* number of garbage collections performed */ + Index max_deg ; /* maximum row degree */ + double default_knobs [COLAMD_KNOBS] ; /* default knobs array */ + + + /* === Check the input arguments ======================================== */ + + if (!stats) + { + COLAMD_DEBUG0 (("colamd: stats not present\n")) ; + return (false) ; + } + for (i = 0 ; i < COLAMD_STATS ; i++) + { + stats [i] = 0 ; + } + stats [COLAMD_STATUS] = COLAMD_OK ; + stats [COLAMD_INFO1] = -1 ; + stats [COLAMD_INFO2] = -1 ; + + if (!A) /* A is not present */ + { + stats [COLAMD_STATUS] = COLAMD_ERROR_A_not_present ; + COLAMD_DEBUG0 (("colamd: A not present\n")) ; + return (false) ; + } + + if (!p) /* p is not present */ + { + stats [COLAMD_STATUS] = COLAMD_ERROR_p_not_present ; + COLAMD_DEBUG0 (("colamd: p not present\n")) ; + return (false) ; + } + + if (n_row < 0) /* n_row must be >= 0 */ + { + stats [COLAMD_STATUS] = COLAMD_ERROR_nrow_negative ; + stats [COLAMD_INFO1] = n_row ; + COLAMD_DEBUG0 (("colamd: nrow negative %d\n", n_row)) ; + return (false) ; + } + + if (n_col < 0) /* n_col must be >= 0 */ + { + stats [COLAMD_STATUS] = COLAMD_ERROR_ncol_negative ; + stats [COLAMD_INFO1] = n_col ; + COLAMD_DEBUG0 (("colamd: ncol negative %d\n", n_col)) ; + return (false) ; + } + + nnz = p [n_col] ; + if (nnz < 0) /* nnz must be >= 0 */ + { + stats [COLAMD_STATUS] = COLAMD_ERROR_nnz_negative ; + stats [COLAMD_INFO1] = nnz ; + COLAMD_DEBUG0 (("colamd: number of entries negative %d\n", nnz)) ; + return (false) ; + } + + if (p [0] != 0) + { + stats [COLAMD_STATUS] = COLAMD_ERROR_p0_nonzero ; + stats [COLAMD_INFO1] = p [0] ; + COLAMD_DEBUG0 (("colamd: p[0] not zero %d\n", p [0])) ; + return (false) ; + } + + /* === If no knobs, set default knobs =================================== */ + + if (!knobs) + { + colamd_set_defaults (default_knobs) ; + knobs = default_knobs ; + } + + /* === Allocate the Row and Col arrays from array A ===================== */ + + Col_size = colamd_c (n_col) ; + Row_size = colamd_r (n_row) ; + need = 2*nnz + n_col + Col_size + Row_size ; + + if (need > Alen) + { + /* not enough space in array A to perform the ordering */ + stats [COLAMD_STATUS] = COLAMD_ERROR_A_too_small ; + stats [COLAMD_INFO1] = need ; + stats [COLAMD_INFO2] = Alen ; + COLAMD_DEBUG0 (("colamd: Need Alen >= %d, given only Alen = %d\n", need,Alen)); + return (false) ; + } + + Alen -= Col_size + Row_size ; + Col = (colamd_col<Index> *) &A [Alen] ; + Row = (Colamd_Row<Index> *) &A [Alen + Col_size] ; + + /* === Construct the row and column data structures ===================== */ + + if (!Eigen::internal::init_rows_cols (n_row, n_col, Row, Col, A, p, stats)) + { + /* input matrix is invalid */ + COLAMD_DEBUG0 (("colamd: Matrix invalid\n")) ; + return (false) ; + } + + /* === Initialize scores, kill dense rows/columns ======================= */ + + Eigen::internal::init_scoring (n_row, n_col, Row, Col, A, p, knobs, + &n_row2, &n_col2, &max_deg) ; + + /* === Order the supercolumns =========================================== */ + + ngarbage = Eigen::internal::find_ordering (n_row, n_col, Alen, Row, Col, A, p, + n_col2, max_deg, 2*nnz) ; + + /* === Order the non-principal columns ================================== */ + + Eigen::internal::order_children (n_col, Col, p) ; + + /* === Return statistics in stats ======================================= */ + + stats [COLAMD_DENSE_ROW] = n_row - n_row2 ; + stats [COLAMD_DENSE_COL] = n_col - n_col2 ; + stats [COLAMD_DEFRAG_COUNT] = ngarbage ; + COLAMD_DEBUG0 (("colamd: done.\n")) ; + return (true) ; +} + +/* ========================================================================== */ +/* === NON-USER-CALLABLE ROUTINES: ========================================== */ +/* ========================================================================== */ + +/* There are no user-callable routines beyond this point in the file */ + + +/* ========================================================================== */ +/* === init_rows_cols ======================================================= */ +/* ========================================================================== */ + +/* + Takes the column form of the matrix in A and creates the row form of the + matrix. Also, row and column attributes are stored in the Col and Row + structs. If the columns are un-sorted or contain duplicate row indices, + this routine will also sort and remove duplicate row indices from the + column form of the matrix. Returns false if the matrix is invalid, + true otherwise. Not user-callable. +*/ +template <typename Index> +static Index init_rows_cols /* returns true if OK, or false otherwise */ + ( + /* === Parameters ======================================================= */ + + Index n_row, /* number of rows of A */ + Index n_col, /* number of columns of A */ + Colamd_Row<Index> Row [], /* of size n_row+1 */ + colamd_col<Index> Col [], /* of size n_col+1 */ + Index A [], /* row indices of A, of size Alen */ + Index p [], /* pointers to columns in A, of size n_col+1 */ + Index stats [COLAMD_STATS] /* colamd statistics */ + ) +{ + /* === Local variables ================================================== */ + + Index col ; /* a column index */ + Index row ; /* a row index */ + Index *cp ; /* a column pointer */ + Index *cp_end ; /* a pointer to the end of a column */ + Index *rp ; /* a row pointer */ + Index *rp_end ; /* a pointer to the end of a row */ + Index last_row ; /* previous row */ + + /* === Initialize columns, and check column pointers ==================== */ + + for (col = 0 ; col < n_col ; col++) + { + Col [col].start = p [col] ; + Col [col].length = p [col+1] - p [col] ; + + if ((Col [col].length) < 0) // extra parentheses to work-around gcc bug 10200 + { + /* column pointers must be non-decreasing */ + stats [COLAMD_STATUS] = COLAMD_ERROR_col_length_negative ; + stats [COLAMD_INFO1] = col ; + stats [COLAMD_INFO2] = Col [col].length ; + COLAMD_DEBUG0 (("colamd: col %d length %d < 0\n", col, Col [col].length)) ; + return (false) ; + } + + Col [col].shared1.thickness = 1 ; + Col [col].shared2.score = 0 ; + Col [col].shared3.prev = COLAMD_EMPTY ; + Col [col].shared4.degree_next = COLAMD_EMPTY ; + } + + /* p [0..n_col] no longer needed, used as "head" in subsequent routines */ + + /* === Scan columns, compute row degrees, and check row indices ========= */ + + stats [COLAMD_INFO3] = 0 ; /* number of duplicate or unsorted row indices*/ + + for (row = 0 ; row < n_row ; row++) + { + Row [row].length = 0 ; + Row [row].shared2.mark = -1 ; + } + + for (col = 0 ; col < n_col ; col++) + { + last_row = -1 ; + + cp = &A [p [col]] ; + cp_end = &A [p [col+1]] ; + + while (cp < cp_end) + { + row = *cp++ ; + + /* make sure row indices within range */ + if (row < 0 || row >= n_row) + { + stats [COLAMD_STATUS] = COLAMD_ERROR_row_index_out_of_bounds ; + stats [COLAMD_INFO1] = col ; + stats [COLAMD_INFO2] = row ; + stats [COLAMD_INFO3] = n_row ; + COLAMD_DEBUG0 (("colamd: row %d col %d out of bounds\n", row, col)) ; + return (false) ; + } + + if (row <= last_row || Row [row].shared2.mark == col) + { + /* row index are unsorted or repeated (or both), thus col */ + /* is jumbled. This is a notice, not an error condition. */ + stats [COLAMD_STATUS] = COLAMD_OK_BUT_JUMBLED ; + stats [COLAMD_INFO1] = col ; + stats [COLAMD_INFO2] = row ; + (stats [COLAMD_INFO3]) ++ ; + COLAMD_DEBUG1 (("colamd: row %d col %d unsorted/duplicate\n",row,col)); + } + + if (Row [row].shared2.mark != col) + { + Row [row].length++ ; + } + else + { + /* this is a repeated entry in the column, */ + /* it will be removed */ + Col [col].length-- ; + } + + /* mark the row as having been seen in this column */ + Row [row].shared2.mark = col ; + + last_row = row ; + } + } + + /* === Compute row pointers ============================================= */ + + /* row form of the matrix starts directly after the column */ + /* form of matrix in A */ + Row [0].start = p [n_col] ; + Row [0].shared1.p = Row [0].start ; + Row [0].shared2.mark = -1 ; + for (row = 1 ; row < n_row ; row++) + { + Row [row].start = Row [row-1].start + Row [row-1].length ; + Row [row].shared1.p = Row [row].start ; + Row [row].shared2.mark = -1 ; + } + + /* === Create row form ================================================== */ + + if (stats [COLAMD_STATUS] == COLAMD_OK_BUT_JUMBLED) + { + /* if cols jumbled, watch for repeated row indices */ + for (col = 0 ; col < n_col ; col++) + { + cp = &A [p [col]] ; + cp_end = &A [p [col+1]] ; + while (cp < cp_end) + { + row = *cp++ ; + if (Row [row].shared2.mark != col) + { + A [(Row [row].shared1.p)++] = col ; + Row [row].shared2.mark = col ; + } + } + } + } + else + { + /* if cols not jumbled, we don't need the mark (this is faster) */ + for (col = 0 ; col < n_col ; col++) + { + cp = &A [p [col]] ; + cp_end = &A [p [col+1]] ; + while (cp < cp_end) + { + A [(Row [*cp++].shared1.p)++] = col ; + } + } + } + + /* === Clear the row marks and set row degrees ========================== */ + + for (row = 0 ; row < n_row ; row++) + { + Row [row].shared2.mark = 0 ; + Row [row].shared1.degree = Row [row].length ; + } + + /* === See if we need to re-create columns ============================== */ + + if (stats [COLAMD_STATUS] == COLAMD_OK_BUT_JUMBLED) + { + COLAMD_DEBUG0 (("colamd: reconstructing column form, matrix jumbled\n")) ; + + + /* === Compute col pointers ========================================= */ + + /* col form of the matrix starts at A [0]. */ + /* Note, we may have a gap between the col form and the row */ + /* form if there were duplicate entries, if so, it will be */ + /* removed upon the first garbage collection */ + Col [0].start = 0 ; + p [0] = Col [0].start ; + for (col = 1 ; col < n_col ; col++) + { + /* note that the lengths here are for pruned columns, i.e. */ + /* no duplicate row indices will exist for these columns */ + Col [col].start = Col [col-1].start + Col [col-1].length ; + p [col] = Col [col].start ; + } + + /* === Re-create col form =========================================== */ + + for (row = 0 ; row < n_row ; row++) + { + rp = &A [Row [row].start] ; + rp_end = rp + Row [row].length ; + while (rp < rp_end) + { + A [(p [*rp++])++] = row ; + } + } + } + + /* === Done. Matrix is not (or no longer) jumbled ====================== */ + + return (true) ; +} + + +/* ========================================================================== */ +/* === init_scoring ========================================================= */ +/* ========================================================================== */ + +/* + Kills dense or empty columns and rows, calculates an initial score for + each column, and places all columns in the degree lists. Not user-callable. +*/ +template <typename Index> +static void init_scoring + ( + /* === Parameters ======================================================= */ + + Index n_row, /* number of rows of A */ + Index n_col, /* number of columns of A */ + Colamd_Row<Index> Row [], /* of size n_row+1 */ + colamd_col<Index> Col [], /* of size n_col+1 */ + Index A [], /* column form and row form of A */ + Index head [], /* of size n_col+1 */ + double knobs [COLAMD_KNOBS],/* parameters */ + Index *p_n_row2, /* number of non-dense, non-empty rows */ + Index *p_n_col2, /* number of non-dense, non-empty columns */ + Index *p_max_deg /* maximum row degree */ + ) +{ + /* === Local variables ================================================== */ + + Index c ; /* a column index */ + Index r, row ; /* a row index */ + Index *cp ; /* a column pointer */ + Index deg ; /* degree of a row or column */ + Index *cp_end ; /* a pointer to the end of a column */ + Index *new_cp ; /* new column pointer */ + Index col_length ; /* length of pruned column */ + Index score ; /* current column score */ + Index n_col2 ; /* number of non-dense, non-empty columns */ + Index n_row2 ; /* number of non-dense, non-empty rows */ + Index dense_row_count ; /* remove rows with more entries than this */ + Index dense_col_count ; /* remove cols with more entries than this */ + Index min_score ; /* smallest column score */ + Index max_deg ; /* maximum row degree */ + Index next_col ; /* Used to add to degree list.*/ + + + /* === Extract knobs ==================================================== */ + + dense_row_count = std::max<Index>(0, (std::min)(Index(knobs [COLAMD_DENSE_ROW] * n_col), n_col)) ; + dense_col_count = std::max<Index>(0, (std::min)(Index(knobs [COLAMD_DENSE_COL] * n_row), n_row)) ; + COLAMD_DEBUG1 (("colamd: densecount: %d %d\n", dense_row_count, dense_col_count)) ; + max_deg = 0 ; + n_col2 = n_col ; + n_row2 = n_row ; + + /* === Kill empty columns =============================================== */ + + /* Put the empty columns at the end in their natural order, so that LU */ + /* factorization can proceed as far as possible. */ + for (c = n_col-1 ; c >= 0 ; c--) + { + deg = Col [c].length ; + if (deg == 0) + { + /* this is a empty column, kill and order it last */ + Col [c].shared2.order = --n_col2 ; + KILL_PRINCIPAL_COL (c) ; + } + } + COLAMD_DEBUG1 (("colamd: null columns killed: %d\n", n_col - n_col2)) ; + + /* === Kill dense columns =============================================== */ + + /* Put the dense columns at the end, in their natural order */ + for (c = n_col-1 ; c >= 0 ; c--) + { + /* skip any dead columns */ + if (COL_IS_DEAD (c)) + { + continue ; + } + deg = Col [c].length ; + if (deg > dense_col_count) + { + /* this is a dense column, kill and order it last */ + Col [c].shared2.order = --n_col2 ; + /* decrement the row degrees */ + cp = &A [Col [c].start] ; + cp_end = cp + Col [c].length ; + while (cp < cp_end) + { + Row [*cp++].shared1.degree-- ; + } + KILL_PRINCIPAL_COL (c) ; + } + } + COLAMD_DEBUG1 (("colamd: Dense and null columns killed: %d\n", n_col - n_col2)) ; + + /* === Kill dense and empty rows ======================================== */ + + for (r = 0 ; r < n_row ; r++) + { + deg = Row [r].shared1.degree ; + COLAMD_ASSERT (deg >= 0 && deg <= n_col) ; + if (deg > dense_row_count || deg == 0) + { + /* kill a dense or empty row */ + KILL_ROW (r) ; + --n_row2 ; + } + else + { + /* keep track of max degree of remaining rows */ + max_deg = (std::max)(max_deg, deg) ; + } + } + COLAMD_DEBUG1 (("colamd: Dense and null rows killed: %d\n", n_row - n_row2)) ; + + /* === Compute initial column scores ==================================== */ + + /* At this point the row degrees are accurate. They reflect the number */ + /* of "live" (non-dense) columns in each row. No empty rows exist. */ + /* Some "live" columns may contain only dead rows, however. These are */ + /* pruned in the code below. */ + + /* now find the initial matlab score for each column */ + for (c = n_col-1 ; c >= 0 ; c--) + { + /* skip dead column */ + if (COL_IS_DEAD (c)) + { + continue ; + } + score = 0 ; + cp = &A [Col [c].start] ; + new_cp = cp ; + cp_end = cp + Col [c].length ; + while (cp < cp_end) + { + /* get a row */ + row = *cp++ ; + /* skip if dead */ + if (ROW_IS_DEAD (row)) + { + continue ; + } + /* compact the column */ + *new_cp++ = row ; + /* add row's external degree */ + score += Row [row].shared1.degree - 1 ; + /* guard against integer overflow */ + score = (std::min)(score, n_col) ; + } + /* determine pruned column length */ + col_length = (Index) (new_cp - &A [Col [c].start]) ; + if (col_length == 0) + { + /* a newly-made null column (all rows in this col are "dense" */ + /* and have already been killed) */ + COLAMD_DEBUG2 (("Newly null killed: %d\n", c)) ; + Col [c].shared2.order = --n_col2 ; + KILL_PRINCIPAL_COL (c) ; + } + else + { + /* set column length and set score */ + COLAMD_ASSERT (score >= 0) ; + COLAMD_ASSERT (score <= n_col) ; + Col [c].length = col_length ; + Col [c].shared2.score = score ; + } + } + COLAMD_DEBUG1 (("colamd: Dense, null, and newly-null columns killed: %d\n", + n_col-n_col2)) ; + + /* At this point, all empty rows and columns are dead. All live columns */ + /* are "clean" (containing no dead rows) and simplicial (no supercolumns */ + /* yet). Rows may contain dead columns, but all live rows contain at */ + /* least one live column. */ + + /* === Initialize degree lists ========================================== */ + + + /* clear the hash buckets */ + for (c = 0 ; c <= n_col ; c++) + { + head [c] = COLAMD_EMPTY ; + } + min_score = n_col ; + /* place in reverse order, so low column indices are at the front */ + /* of the lists. This is to encourage natural tie-breaking */ + for (c = n_col-1 ; c >= 0 ; c--) + { + /* only add principal columns to degree lists */ + if (COL_IS_ALIVE (c)) + { + COLAMD_DEBUG4 (("place %d score %d minscore %d ncol %d\n", + c, Col [c].shared2.score, min_score, n_col)) ; + + /* === Add columns score to DList =============================== */ + + score = Col [c].shared2.score ; + + COLAMD_ASSERT (min_score >= 0) ; + COLAMD_ASSERT (min_score <= n_col) ; + COLAMD_ASSERT (score >= 0) ; + COLAMD_ASSERT (score <= n_col) ; + COLAMD_ASSERT (head [score] >= COLAMD_EMPTY) ; + + /* now add this column to dList at proper score location */ + next_col = head [score] ; + Col [c].shared3.prev = COLAMD_EMPTY ; + Col [c].shared4.degree_next = next_col ; + + /* if there already was a column with the same score, set its */ + /* previous pointer to this new column */ + if (next_col != COLAMD_EMPTY) + { + Col [next_col].shared3.prev = c ; + } + head [score] = c ; + + /* see if this score is less than current min */ + min_score = (std::min)(min_score, score) ; + + + } + } + + + /* === Return number of remaining columns, and max row degree =========== */ + + *p_n_col2 = n_col2 ; + *p_n_row2 = n_row2 ; + *p_max_deg = max_deg ; +} + + +/* ========================================================================== */ +/* === find_ordering ======================================================== */ +/* ========================================================================== */ + +/* + Order the principal columns of the supercolumn form of the matrix + (no supercolumns on input). Uses a minimum approximate column minimum + degree ordering method. Not user-callable. +*/ +template <typename Index> +static Index find_ordering /* return the number of garbage collections */ + ( + /* === Parameters ======================================================= */ + + Index n_row, /* number of rows of A */ + Index n_col, /* number of columns of A */ + Index Alen, /* size of A, 2*nnz + n_col or larger */ + Colamd_Row<Index> Row [], /* of size n_row+1 */ + colamd_col<Index> Col [], /* of size n_col+1 */ + Index A [], /* column form and row form of A */ + Index head [], /* of size n_col+1 */ + Index n_col2, /* Remaining columns to order */ + Index max_deg, /* Maximum row degree */ + Index pfree /* index of first free slot (2*nnz on entry) */ + ) +{ + /* === Local variables ================================================== */ + + Index k ; /* current pivot ordering step */ + Index pivot_col ; /* current pivot column */ + Index *cp ; /* a column pointer */ + Index *rp ; /* a row pointer */ + Index pivot_row ; /* current pivot row */ + Index *new_cp ; /* modified column pointer */ + Index *new_rp ; /* modified row pointer */ + Index pivot_row_start ; /* pointer to start of pivot row */ + Index pivot_row_degree ; /* number of columns in pivot row */ + Index pivot_row_length ; /* number of supercolumns in pivot row */ + Index pivot_col_score ; /* score of pivot column */ + Index needed_memory ; /* free space needed for pivot row */ + Index *cp_end ; /* pointer to the end of a column */ + Index *rp_end ; /* pointer to the end of a row */ + Index row ; /* a row index */ + Index col ; /* a column index */ + Index max_score ; /* maximum possible score */ + Index cur_score ; /* score of current column */ + unsigned int hash ; /* hash value for supernode detection */ + Index head_column ; /* head of hash bucket */ + Index first_col ; /* first column in hash bucket */ + Index tag_mark ; /* marker value for mark array */ + Index row_mark ; /* Row [row].shared2.mark */ + Index set_difference ; /* set difference size of row with pivot row */ + Index min_score ; /* smallest column score */ + Index col_thickness ; /* "thickness" (no. of columns in a supercol) */ + Index max_mark ; /* maximum value of tag_mark */ + Index pivot_col_thickness ; /* number of columns represented by pivot col */ + Index prev_col ; /* Used by Dlist operations. */ + Index next_col ; /* Used by Dlist operations. */ + Index ngarbage ; /* number of garbage collections performed */ + + + /* === Initialization and clear mark ==================================== */ + + max_mark = INT_MAX - n_col ; /* INT_MAX defined in <limits.h> */ + tag_mark = Eigen::internal::clear_mark (n_row, Row) ; + min_score = 0 ; + ngarbage = 0 ; + COLAMD_DEBUG1 (("colamd: Ordering, n_col2=%d\n", n_col2)) ; + + /* === Order the columns ================================================ */ + + for (k = 0 ; k < n_col2 ; /* 'k' is incremented below */) + { + + /* === Select pivot column, and order it ============================ */ + + /* make sure degree list isn't empty */ + COLAMD_ASSERT (min_score >= 0) ; + COLAMD_ASSERT (min_score <= n_col) ; + COLAMD_ASSERT (head [min_score] >= COLAMD_EMPTY) ; + + /* get pivot column from head of minimum degree list */ + while (head [min_score] == COLAMD_EMPTY && min_score < n_col) + { + min_score++ ; + } + pivot_col = head [min_score] ; + COLAMD_ASSERT (pivot_col >= 0 && pivot_col <= n_col) ; + next_col = Col [pivot_col].shared4.degree_next ; + head [min_score] = next_col ; + if (next_col != COLAMD_EMPTY) + { + Col [next_col].shared3.prev = COLAMD_EMPTY ; + } + + COLAMD_ASSERT (COL_IS_ALIVE (pivot_col)) ; + COLAMD_DEBUG3 (("Pivot col: %d\n", pivot_col)) ; + + /* remember score for defrag check */ + pivot_col_score = Col [pivot_col].shared2.score ; + + /* the pivot column is the kth column in the pivot order */ + Col [pivot_col].shared2.order = k ; + + /* increment order count by column thickness */ + pivot_col_thickness = Col [pivot_col].shared1.thickness ; + k += pivot_col_thickness ; + COLAMD_ASSERT (pivot_col_thickness > 0) ; + + /* === Garbage_collection, if necessary ============================= */ + + needed_memory = (std::min)(pivot_col_score, n_col - k) ; + if (pfree + needed_memory >= Alen) + { + pfree = Eigen::internal::garbage_collection (n_row, n_col, Row, Col, A, &A [pfree]) ; + ngarbage++ ; + /* after garbage collection we will have enough */ + COLAMD_ASSERT (pfree + needed_memory < Alen) ; + /* garbage collection has wiped out the Row[].shared2.mark array */ + tag_mark = Eigen::internal::clear_mark (n_row, Row) ; + + } + + /* === Compute pivot row pattern ==================================== */ + + /* get starting location for this new merged row */ + pivot_row_start = pfree ; + + /* initialize new row counts to zero */ + pivot_row_degree = 0 ; + + /* tag pivot column as having been visited so it isn't included */ + /* in merged pivot row */ + Col [pivot_col].shared1.thickness = -pivot_col_thickness ; + + /* pivot row is the union of all rows in the pivot column pattern */ + cp = &A [Col [pivot_col].start] ; + cp_end = cp + Col [pivot_col].length ; + while (cp < cp_end) + { + /* get a row */ + row = *cp++ ; + COLAMD_DEBUG4 (("Pivot col pattern %d %d\n", ROW_IS_ALIVE (row), row)) ; + /* skip if row is dead */ + if (ROW_IS_DEAD (row)) + { + continue ; + } + rp = &A [Row [row].start] ; + rp_end = rp + Row [row].length ; + while (rp < rp_end) + { + /* get a column */ + col = *rp++ ; + /* add the column, if alive and untagged */ + col_thickness = Col [col].shared1.thickness ; + if (col_thickness > 0 && COL_IS_ALIVE (col)) + { + /* tag column in pivot row */ + Col [col].shared1.thickness = -col_thickness ; + COLAMD_ASSERT (pfree < Alen) ; + /* place column in pivot row */ + A [pfree++] = col ; + pivot_row_degree += col_thickness ; + } + } + } + + /* clear tag on pivot column */ + Col [pivot_col].shared1.thickness = pivot_col_thickness ; + max_deg = (std::max)(max_deg, pivot_row_degree) ; + + + /* === Kill all rows used to construct pivot row ==================== */ + + /* also kill pivot row, temporarily */ + cp = &A [Col [pivot_col].start] ; + cp_end = cp + Col [pivot_col].length ; + while (cp < cp_end) + { + /* may be killing an already dead row */ + row = *cp++ ; + COLAMD_DEBUG3 (("Kill row in pivot col: %d\n", row)) ; + KILL_ROW (row) ; + } + + /* === Select a row index to use as the new pivot row =============== */ + + pivot_row_length = pfree - pivot_row_start ; + if (pivot_row_length > 0) + { + /* pick the "pivot" row arbitrarily (first row in col) */ + pivot_row = A [Col [pivot_col].start] ; + COLAMD_DEBUG3 (("Pivotal row is %d\n", pivot_row)) ; + } + else + { + /* there is no pivot row, since it is of zero length */ + pivot_row = COLAMD_EMPTY ; + COLAMD_ASSERT (pivot_row_length == 0) ; + } + COLAMD_ASSERT (Col [pivot_col].length > 0 || pivot_row_length == 0) ; + + /* === Approximate degree computation =============================== */ + + /* Here begins the computation of the approximate degree. The column */ + /* score is the sum of the pivot row "length", plus the size of the */ + /* set differences of each row in the column minus the pattern of the */ + /* pivot row itself. The column ("thickness") itself is also */ + /* excluded from the column score (we thus use an approximate */ + /* external degree). */ + + /* The time taken by the following code (compute set differences, and */ + /* add them up) is proportional to the size of the data structure */ + /* being scanned - that is, the sum of the sizes of each column in */ + /* the pivot row. Thus, the amortized time to compute a column score */ + /* is proportional to the size of that column (where size, in this */ + /* context, is the column "length", or the number of row indices */ + /* in that column). The number of row indices in a column is */ + /* monotonically non-decreasing, from the length of the original */ + /* column on input to colamd. */ + + /* === Compute set differences ====================================== */ + + COLAMD_DEBUG3 (("** Computing set differences phase. **\n")) ; + + /* pivot row is currently dead - it will be revived later. */ + + COLAMD_DEBUG3 (("Pivot row: ")) ; + /* for each column in pivot row */ + rp = &A [pivot_row_start] ; + rp_end = rp + pivot_row_length ; + while (rp < rp_end) + { + col = *rp++ ; + COLAMD_ASSERT (COL_IS_ALIVE (col) && col != pivot_col) ; + COLAMD_DEBUG3 (("Col: %d\n", col)) ; + + /* clear tags used to construct pivot row pattern */ + col_thickness = -Col [col].shared1.thickness ; + COLAMD_ASSERT (col_thickness > 0) ; + Col [col].shared1.thickness = col_thickness ; + + /* === Remove column from degree list =========================== */ + + cur_score = Col [col].shared2.score ; + prev_col = Col [col].shared3.prev ; + next_col = Col [col].shared4.degree_next ; + COLAMD_ASSERT (cur_score >= 0) ; + COLAMD_ASSERT (cur_score <= n_col) ; + COLAMD_ASSERT (cur_score >= COLAMD_EMPTY) ; + if (prev_col == COLAMD_EMPTY) + { + head [cur_score] = next_col ; + } + else + { + Col [prev_col].shared4.degree_next = next_col ; + } + if (next_col != COLAMD_EMPTY) + { + Col [next_col].shared3.prev = prev_col ; + } + + /* === Scan the column ========================================== */ + + cp = &A [Col [col].start] ; + cp_end = cp + Col [col].length ; + while (cp < cp_end) + { + /* get a row */ + row = *cp++ ; + row_mark = Row [row].shared2.mark ; + /* skip if dead */ + if (ROW_IS_MARKED_DEAD (row_mark)) + { + continue ; + } + COLAMD_ASSERT (row != pivot_row) ; + set_difference = row_mark - tag_mark ; + /* check if the row has been seen yet */ + if (set_difference < 0) + { + COLAMD_ASSERT (Row [row].shared1.degree <= max_deg) ; + set_difference = Row [row].shared1.degree ; + } + /* subtract column thickness from this row's set difference */ + set_difference -= col_thickness ; + COLAMD_ASSERT (set_difference >= 0) ; + /* absorb this row if the set difference becomes zero */ + if (set_difference == 0) + { + COLAMD_DEBUG3 (("aggressive absorption. Row: %d\n", row)) ; + KILL_ROW (row) ; + } + else + { + /* save the new mark */ + Row [row].shared2.mark = set_difference + tag_mark ; + } + } + } + + + /* === Add up set differences for each column ======================= */ + + COLAMD_DEBUG3 (("** Adding set differences phase. **\n")) ; + + /* for each column in pivot row */ + rp = &A [pivot_row_start] ; + rp_end = rp + pivot_row_length ; + while (rp < rp_end) + { + /* get a column */ + col = *rp++ ; + COLAMD_ASSERT (COL_IS_ALIVE (col) && col != pivot_col) ; + hash = 0 ; + cur_score = 0 ; + cp = &A [Col [col].start] ; + /* compact the column */ + new_cp = cp ; + cp_end = cp + Col [col].length ; + + COLAMD_DEBUG4 (("Adding set diffs for Col: %d.\n", col)) ; + + while (cp < cp_end) + { + /* get a row */ + row = *cp++ ; + COLAMD_ASSERT(row >= 0 && row < n_row) ; + row_mark = Row [row].shared2.mark ; + /* skip if dead */ + if (ROW_IS_MARKED_DEAD (row_mark)) + { + continue ; + } + COLAMD_ASSERT (row_mark > tag_mark) ; + /* compact the column */ + *new_cp++ = row ; + /* compute hash function */ + hash += row ; + /* add set difference */ + cur_score += row_mark - tag_mark ; + /* integer overflow... */ + cur_score = (std::min)(cur_score, n_col) ; + } + + /* recompute the column's length */ + Col [col].length = (Index) (new_cp - &A [Col [col].start]) ; + + /* === Further mass elimination ================================= */ + + if (Col [col].length == 0) + { + COLAMD_DEBUG4 (("further mass elimination. Col: %d\n", col)) ; + /* nothing left but the pivot row in this column */ + KILL_PRINCIPAL_COL (col) ; + pivot_row_degree -= Col [col].shared1.thickness ; + COLAMD_ASSERT (pivot_row_degree >= 0) ; + /* order it */ + Col [col].shared2.order = k ; + /* increment order count by column thickness */ + k += Col [col].shared1.thickness ; + } + else + { + /* === Prepare for supercolumn detection ==================== */ + + COLAMD_DEBUG4 (("Preparing supercol detection for Col: %d.\n", col)) ; + + /* save score so far */ + Col [col].shared2.score = cur_score ; + + /* add column to hash table, for supercolumn detection */ + hash %= n_col + 1 ; + + COLAMD_DEBUG4 ((" Hash = %d, n_col = %d.\n", hash, n_col)) ; + COLAMD_ASSERT (hash <= n_col) ; + + head_column = head [hash] ; + if (head_column > COLAMD_EMPTY) + { + /* degree list "hash" is non-empty, use prev (shared3) of */ + /* first column in degree list as head of hash bucket */ + first_col = Col [head_column].shared3.headhash ; + Col [head_column].shared3.headhash = col ; + } + else + { + /* degree list "hash" is empty, use head as hash bucket */ + first_col = - (head_column + 2) ; + head [hash] = - (col + 2) ; + } + Col [col].shared4.hash_next = first_col ; + + /* save hash function in Col [col].shared3.hash */ + Col [col].shared3.hash = (Index) hash ; + COLAMD_ASSERT (COL_IS_ALIVE (col)) ; + } + } + + /* The approximate external column degree is now computed. */ + + /* === Supercolumn detection ======================================== */ + + COLAMD_DEBUG3 (("** Supercolumn detection phase. **\n")) ; + + Eigen::internal::detect_super_cols (Col, A, head, pivot_row_start, pivot_row_length) ; + + /* === Kill the pivotal column ====================================== */ + + KILL_PRINCIPAL_COL (pivot_col) ; + + /* === Clear mark =================================================== */ + + tag_mark += (max_deg + 1) ; + if (tag_mark >= max_mark) + { + COLAMD_DEBUG2 (("clearing tag_mark\n")) ; + tag_mark = Eigen::internal::clear_mark (n_row, Row) ; + } + + /* === Finalize the new pivot row, and column scores ================ */ + + COLAMD_DEBUG3 (("** Finalize scores phase. **\n")) ; + + /* for each column in pivot row */ + rp = &A [pivot_row_start] ; + /* compact the pivot row */ + new_rp = rp ; + rp_end = rp + pivot_row_length ; + while (rp < rp_end) + { + col = *rp++ ; + /* skip dead columns */ + if (COL_IS_DEAD (col)) + { + continue ; + } + *new_rp++ = col ; + /* add new pivot row to column */ + A [Col [col].start + (Col [col].length++)] = pivot_row ; + + /* retrieve score so far and add on pivot row's degree. */ + /* (we wait until here for this in case the pivot */ + /* row's degree was reduced due to mass elimination). */ + cur_score = Col [col].shared2.score + pivot_row_degree ; + + /* calculate the max possible score as the number of */ + /* external columns minus the 'k' value minus the */ + /* columns thickness */ + max_score = n_col - k - Col [col].shared1.thickness ; + + /* make the score the external degree of the union-of-rows */ + cur_score -= Col [col].shared1.thickness ; + + /* make sure score is less or equal than the max score */ + cur_score = (std::min)(cur_score, max_score) ; + COLAMD_ASSERT (cur_score >= 0) ; + + /* store updated score */ + Col [col].shared2.score = cur_score ; + + /* === Place column back in degree list ========================= */ + + COLAMD_ASSERT (min_score >= 0) ; + COLAMD_ASSERT (min_score <= n_col) ; + COLAMD_ASSERT (cur_score >= 0) ; + COLAMD_ASSERT (cur_score <= n_col) ; + COLAMD_ASSERT (head [cur_score] >= COLAMD_EMPTY) ; + next_col = head [cur_score] ; + Col [col].shared4.degree_next = next_col ; + Col [col].shared3.prev = COLAMD_EMPTY ; + if (next_col != COLAMD_EMPTY) + { + Col [next_col].shared3.prev = col ; + } + head [cur_score] = col ; + + /* see if this score is less than current min */ + min_score = (std::min)(min_score, cur_score) ; + + } + + /* === Resurrect the new pivot row ================================== */ + + if (pivot_row_degree > 0) + { + /* update pivot row length to reflect any cols that were killed */ + /* during super-col detection and mass elimination */ + Row [pivot_row].start = pivot_row_start ; + Row [pivot_row].length = (Index) (new_rp - &A[pivot_row_start]) ; + Row [pivot_row].shared1.degree = pivot_row_degree ; + Row [pivot_row].shared2.mark = 0 ; + /* pivot row is no longer dead */ + } + } + + /* === All principal columns have now been ordered ====================== */ + + return (ngarbage) ; +} + + +/* ========================================================================== */ +/* === order_children ======================================================= */ +/* ========================================================================== */ + +/* + The find_ordering routine has ordered all of the principal columns (the + representatives of the supercolumns). The non-principal columns have not + yet been ordered. This routine orders those columns by walking up the + parent tree (a column is a child of the column which absorbed it). The + final permutation vector is then placed in p [0 ... n_col-1], with p [0] + being the first column, and p [n_col-1] being the last. It doesn't look + like it at first glance, but be assured that this routine takes time linear + in the number of columns. Although not immediately obvious, the time + taken by this routine is O (n_col), that is, linear in the number of + columns. Not user-callable. +*/ +template <typename Index> +static inline void order_children +( + /* === Parameters ======================================================= */ + + Index n_col, /* number of columns of A */ + colamd_col<Index> Col [], /* of size n_col+1 */ + Index p [] /* p [0 ... n_col-1] is the column permutation*/ + ) +{ + /* === Local variables ================================================== */ + + Index i ; /* loop counter for all columns */ + Index c ; /* column index */ + Index parent ; /* index of column's parent */ + Index order ; /* column's order */ + + /* === Order each non-principal column ================================== */ + + for (i = 0 ; i < n_col ; i++) + { + /* find an un-ordered non-principal column */ + COLAMD_ASSERT (COL_IS_DEAD (i)) ; + if (!COL_IS_DEAD_PRINCIPAL (i) && Col [i].shared2.order == COLAMD_EMPTY) + { + parent = i ; + /* once found, find its principal parent */ + do + { + parent = Col [parent].shared1.parent ; + } while (!COL_IS_DEAD_PRINCIPAL (parent)) ; + + /* now, order all un-ordered non-principal columns along path */ + /* to this parent. collapse tree at the same time */ + c = i ; + /* get order of parent */ + order = Col [parent].shared2.order ; + + do + { + COLAMD_ASSERT (Col [c].shared2.order == COLAMD_EMPTY) ; + + /* order this column */ + Col [c].shared2.order = order++ ; + /* collaps tree */ + Col [c].shared1.parent = parent ; + + /* get immediate parent of this column */ + c = Col [c].shared1.parent ; + + /* continue until we hit an ordered column. There are */ + /* guarranteed not to be anymore unordered columns */ + /* above an ordered column */ + } while (Col [c].shared2.order == COLAMD_EMPTY) ; + + /* re-order the super_col parent to largest order for this group */ + Col [parent].shared2.order = order ; + } + } + + /* === Generate the permutation ========================================= */ + + for (c = 0 ; c < n_col ; c++) + { + p [Col [c].shared2.order] = c ; + } +} + + +/* ========================================================================== */ +/* === detect_super_cols ==================================================== */ +/* ========================================================================== */ + +/* + Detects supercolumns by finding matches between columns in the hash buckets. + Check amongst columns in the set A [row_start ... row_start + row_length-1]. + The columns under consideration are currently *not* in the degree lists, + and have already been placed in the hash buckets. + + The hash bucket for columns whose hash function is equal to h is stored + as follows: + + if head [h] is >= 0, then head [h] contains a degree list, so: + + head [h] is the first column in degree bucket h. + Col [head [h]].headhash gives the first column in hash bucket h. + + otherwise, the degree list is empty, and: + + -(head [h] + 2) is the first column in hash bucket h. + + For a column c in a hash bucket, Col [c].shared3.prev is NOT a "previous + column" pointer. Col [c].shared3.hash is used instead as the hash number + for that column. The value of Col [c].shared4.hash_next is the next column + in the same hash bucket. + + Assuming no, or "few" hash collisions, the time taken by this routine is + linear in the sum of the sizes (lengths) of each column whose score has + just been computed in the approximate degree computation. + Not user-callable. +*/ +template <typename Index> +static void detect_super_cols +( + /* === Parameters ======================================================= */ + + colamd_col<Index> Col [], /* of size n_col+1 */ + Index A [], /* row indices of A */ + Index head [], /* head of degree lists and hash buckets */ + Index row_start, /* pointer to set of columns to check */ + Index row_length /* number of columns to check */ +) +{ + /* === Local variables ================================================== */ + + Index hash ; /* hash value for a column */ + Index *rp ; /* pointer to a row */ + Index c ; /* a column index */ + Index super_c ; /* column index of the column to absorb into */ + Index *cp1 ; /* column pointer for column super_c */ + Index *cp2 ; /* column pointer for column c */ + Index length ; /* length of column super_c */ + Index prev_c ; /* column preceding c in hash bucket */ + Index i ; /* loop counter */ + Index *rp_end ; /* pointer to the end of the row */ + Index col ; /* a column index in the row to check */ + Index head_column ; /* first column in hash bucket or degree list */ + Index first_col ; /* first column in hash bucket */ + + /* === Consider each column in the row ================================== */ + + rp = &A [row_start] ; + rp_end = rp + row_length ; + while (rp < rp_end) + { + col = *rp++ ; + if (COL_IS_DEAD (col)) + { + continue ; + } + + /* get hash number for this column */ + hash = Col [col].shared3.hash ; + COLAMD_ASSERT (hash <= n_col) ; + + /* === Get the first column in this hash bucket ===================== */ + + head_column = head [hash] ; + if (head_column > COLAMD_EMPTY) + { + first_col = Col [head_column].shared3.headhash ; + } + else + { + first_col = - (head_column + 2) ; + } + + /* === Consider each column in the hash bucket ====================== */ + + for (super_c = first_col ; super_c != COLAMD_EMPTY ; + super_c = Col [super_c].shared4.hash_next) + { + COLAMD_ASSERT (COL_IS_ALIVE (super_c)) ; + COLAMD_ASSERT (Col [super_c].shared3.hash == hash) ; + length = Col [super_c].length ; + + /* prev_c is the column preceding column c in the hash bucket */ + prev_c = super_c ; + + /* === Compare super_c with all columns after it ================ */ + + for (c = Col [super_c].shared4.hash_next ; + c != COLAMD_EMPTY ; c = Col [c].shared4.hash_next) + { + COLAMD_ASSERT (c != super_c) ; + COLAMD_ASSERT (COL_IS_ALIVE (c)) ; + COLAMD_ASSERT (Col [c].shared3.hash == hash) ; + + /* not identical if lengths or scores are different */ + if (Col [c].length != length || + Col [c].shared2.score != Col [super_c].shared2.score) + { + prev_c = c ; + continue ; + } + + /* compare the two columns */ + cp1 = &A [Col [super_c].start] ; + cp2 = &A [Col [c].start] ; + + for (i = 0 ; i < length ; i++) + { + /* the columns are "clean" (no dead rows) */ + COLAMD_ASSERT (ROW_IS_ALIVE (*cp1)) ; + COLAMD_ASSERT (ROW_IS_ALIVE (*cp2)) ; + /* row indices will same order for both supercols, */ + /* no gather scatter nessasary */ + if (*cp1++ != *cp2++) + { + break ; + } + } + + /* the two columns are different if the for-loop "broke" */ + if (i != length) + { + prev_c = c ; + continue ; + } + + /* === Got it! two columns are identical =================== */ + + COLAMD_ASSERT (Col [c].shared2.score == Col [super_c].shared2.score) ; + + Col [super_c].shared1.thickness += Col [c].shared1.thickness ; + Col [c].shared1.parent = super_c ; + KILL_NON_PRINCIPAL_COL (c) ; + /* order c later, in order_children() */ + Col [c].shared2.order = COLAMD_EMPTY ; + /* remove c from hash bucket */ + Col [prev_c].shared4.hash_next = Col [c].shared4.hash_next ; + } + } + + /* === Empty this hash bucket ======================================= */ + + if (head_column > COLAMD_EMPTY) + { + /* corresponding degree list "hash" is not empty */ + Col [head_column].shared3.headhash = COLAMD_EMPTY ; + } + else + { + /* corresponding degree list "hash" is empty */ + head [hash] = COLAMD_EMPTY ; + } + } +} + + +/* ========================================================================== */ +/* === garbage_collection =================================================== */ +/* ========================================================================== */ + +/* + Defragments and compacts columns and rows in the workspace A. Used when + all avaliable memory has been used while performing row merging. Returns + the index of the first free position in A, after garbage collection. The + time taken by this routine is linear is the size of the array A, which is + itself linear in the number of nonzeros in the input matrix. + Not user-callable. +*/ +template <typename Index> +static Index garbage_collection /* returns the new value of pfree */ + ( + /* === Parameters ======================================================= */ + + Index n_row, /* number of rows */ + Index n_col, /* number of columns */ + Colamd_Row<Index> Row [], /* row info */ + colamd_col<Index> Col [], /* column info */ + Index A [], /* A [0 ... Alen-1] holds the matrix */ + Index *pfree /* &A [0] ... pfree is in use */ + ) +{ + /* === Local variables ================================================== */ + + Index *psrc ; /* source pointer */ + Index *pdest ; /* destination pointer */ + Index j ; /* counter */ + Index r ; /* a row index */ + Index c ; /* a column index */ + Index length ; /* length of a row or column */ + + /* === Defragment the columns =========================================== */ + + pdest = &A[0] ; + for (c = 0 ; c < n_col ; c++) + { + if (COL_IS_ALIVE (c)) + { + psrc = &A [Col [c].start] ; + + /* move and compact the column */ + COLAMD_ASSERT (pdest <= psrc) ; + Col [c].start = (Index) (pdest - &A [0]) ; + length = Col [c].length ; + for (j = 0 ; j < length ; j++) + { + r = *psrc++ ; + if (ROW_IS_ALIVE (r)) + { + *pdest++ = r ; + } + } + Col [c].length = (Index) (pdest - &A [Col [c].start]) ; + } + } + + /* === Prepare to defragment the rows =================================== */ + + for (r = 0 ; r < n_row ; r++) + { + if (ROW_IS_ALIVE (r)) + { + if (Row [r].length == 0) + { + /* this row is of zero length. cannot compact it, so kill it */ + COLAMD_DEBUG3 (("Defrag row kill\n")) ; + KILL_ROW (r) ; + } + else + { + /* save first column index in Row [r].shared2.first_column */ + psrc = &A [Row [r].start] ; + Row [r].shared2.first_column = *psrc ; + COLAMD_ASSERT (ROW_IS_ALIVE (r)) ; + /* flag the start of the row with the one's complement of row */ + *psrc = ONES_COMPLEMENT (r) ; + + } + } + } + + /* === Defragment the rows ============================================== */ + + psrc = pdest ; + while (psrc < pfree) + { + /* find a negative number ... the start of a row */ + if (*psrc++ < 0) + { + psrc-- ; + /* get the row index */ + r = ONES_COMPLEMENT (*psrc) ; + COLAMD_ASSERT (r >= 0 && r < n_row) ; + /* restore first column index */ + *psrc = Row [r].shared2.first_column ; + COLAMD_ASSERT (ROW_IS_ALIVE (r)) ; + + /* move and compact the row */ + COLAMD_ASSERT (pdest <= psrc) ; + Row [r].start = (Index) (pdest - &A [0]) ; + length = Row [r].length ; + for (j = 0 ; j < length ; j++) + { + c = *psrc++ ; + if (COL_IS_ALIVE (c)) + { + *pdest++ = c ; + } + } + Row [r].length = (Index) (pdest - &A [Row [r].start]) ; + + } + } + /* ensure we found all the rows */ + COLAMD_ASSERT (debug_rows == 0) ; + + /* === Return the new value of pfree ==================================== */ + + return ((Index) (pdest - &A [0])) ; +} + + +/* ========================================================================== */ +/* === clear_mark =========================================================== */ +/* ========================================================================== */ + +/* + Clears the Row [].shared2.mark array, and returns the new tag_mark. + Return value is the new tag_mark. Not user-callable. +*/ +template <typename Index> +static inline Index clear_mark /* return the new value for tag_mark */ + ( + /* === Parameters ======================================================= */ + + Index n_row, /* number of rows in A */ + Colamd_Row<Index> Row [] /* Row [0 ... n_row-1].shared2.mark is set to zero */ + ) +{ + /* === Local variables ================================================== */ + + Index r ; + + for (r = 0 ; r < n_row ; r++) + { + if (ROW_IS_ALIVE (r)) + { + Row [r].shared2.mark = 0 ; + } + } + return (1) ; +} + + +} // namespace internal +#endif |