"use strict";
Object.defineProperty(exports, "__esModule", {
value: true
});
exports.createCsAmd = void 0;
var _factory = require("../../../utils/factory.js");
var _csFkeep = require("./csFkeep.js");
var _csFlip = require("./csFlip.js");
var _csTdfs = require("./csTdfs.js");
// Copyright (c) 2006-2024, Timothy A. Davis, All Rights Reserved.
// SPDX-License-Identifier: LGPL-2.1+
// https://github.com/DrTimothyAldenDavis/SuiteSparse/tree/dev/CSparse/Source
const name = 'csAmd';
const dependencies = ['add', 'multiply', 'transpose'];
const createCsAmd = exports.createCsAmd = /* #__PURE__ */(0, _factory.factory)(name, dependencies, _ref => {
let {
add,
multiply,
transpose
} = _ref;
/**
* Approximate minimum degree ordering. The minimum degree algorithm is a widely used
* heuristic for finding a permutation P so that P*A*P' has fewer nonzeros in its factorization
* than A. It is a gready method that selects the sparsest pivot row and column during the course
* of a right looking sparse Cholesky factorization.
*
* @param {Number} order 0: Natural, 1: Cholesky, 2: LU, 3: QR
* @param {Matrix} m Sparse Matrix
*/
return function csAmd(order, a) {
// check input parameters
if (!a || order <= 0 || order > 3) {
return null;
}
// a matrix arrays
const asize = a._size;
// rows and columns
const m = asize[0];
const n = asize[1];
// initialize vars
let lemax = 0;
// dense threshold
let dense = Math.max(16, 10 * Math.sqrt(n));
dense = Math.min(n - 2, dense);
// create target matrix C
const cm = _createTargetMatrix(order, a, m, n, dense);
// drop diagonal entries
(0, _csFkeep.csFkeep)(cm, _diag, null);
// C matrix arrays
const cindex = cm._index;
const cptr = cm._ptr;
// number of nonzero elements in C
let cnz = cptr[n];
// allocate result (n+1)
const P = [];
// create workspace (8 * (n + 1))
const W = [];
const len = 0; // first n + 1 entries
const nv = n + 1; // next n + 1 entries
const next = 2 * (n + 1); // next n + 1 entries
const head = 3 * (n + 1); // next n + 1 entries
const elen = 4 * (n + 1); // next n + 1 entries
const degree = 5 * (n + 1); // next n + 1 entries
const w = 6 * (n + 1); // next n + 1 entries
const hhead = 7 * (n + 1); // last n + 1 entries
// use P as workspace for last
const last = P;
// initialize quotient graph
let mark = _initializeQuotientGraph(n, cptr, W, len, head, last, next, hhead, nv, w, elen, degree);
// initialize degree lists
let nel = _initializeDegreeLists(n, cptr, W, degree, elen, w, dense, nv, head, last, next);
// minimum degree node
let mindeg = 0;
// vars
let i, j, k, k1, k2, e, pj, ln, nvi, pk, eln, p1, p2, pn, h, d;
// while (selecting pivots) do
while (nel < n) {
// select node of minimum approximate degree. amd() is now ready to start eliminating the graph. It first
// finds a node k of minimum degree and removes it from its degree list. The variable nel keeps track of thow
// many nodes have been eliminated.
for (k = -1; mindeg < n && (k = W[head + mindeg]) === -1; mindeg++);
if (W[next + k] !== -1) {
last[W[next + k]] = -1;
}
// remove k from degree list
W[head + mindeg] = W[next + k];
// elenk = |Ek|
const elenk = W[elen + k];
// # of nodes k represents
let nvk = W[nv + k];
// W[nv + k] nodes of A eliminated
nel += nvk;
// Construct a new element. The new element Lk is constructed in place if |Ek| = 0. nv[i] is
// negated for all nodes i in Lk to flag them as members of this set. Each node i is removed from the
// degree lists. All elements e in Ek are absorved into element k.
let dk = 0;
// flag k as in Lk
W[nv + k] = -nvk;
let p = cptr[k];
// do in place if W[elen + k] === 0
const pk1 = elenk === 0 ? p : cnz;
let pk2 = pk1;
for (k1 = 1; k1 <= elenk + 1; k1++) {
if (k1 > elenk) {
// search the nodes in k
e = k;
// list of nodes starts at cindex[pj]
pj = p;
// length of list of nodes in k
ln = W[len + k] - elenk;
} else {
// search the nodes in e
e = cindex[p++];
pj = cptr[e];
// length of list of nodes in e
ln = W[len + e];
}
for (k2 = 1; k2 <= ln; k2++) {
i = cindex[pj++];
// check node i dead, or seen
if ((nvi = W[nv + i]) <= 0) {
continue;
}
// W[degree + Lk] += size of node i
dk += nvi;
// negate W[nv + i] to denote i in Lk
W[nv + i] = -nvi;
// place i in Lk
cindex[pk2++] = i;
if (W[next + i] !== -1) {
last[W[next + i]] = last[i];
}
// check we need to remove i from degree list
if (last[i] !== -1) {
W[next + last[i]] = W[next + i];
} else {
W[head + W[degree + i]] = W[next + i];
}
}
if (e !== k) {
// absorb e into k
cptr[e] = (0, _csFlip.csFlip)(k);
// e is now a dead element
W[w + e] = 0;
}
}
// cindex[cnz...nzmax] is free
if (elenk !== 0) {
cnz = pk2;
}
// external degree of k - |Lk\i|
W[degree + k] = dk;
// element k is in cindex[pk1..pk2-1]
cptr[k] = pk1;
W[len + k] = pk2 - pk1;
// k is now an element
W[elen + k] = -2;
// Find set differences. The scan1 function now computes the set differences |Le \ Lk| for all elements e. At the start of the
// scan, no entry in the w array is greater than or equal to mark.
// clear w if necessary
mark = _wclear(mark, lemax, W, w, n);
// scan 1: find |Le\Lk|
for (pk = pk1; pk < pk2; pk++) {
i = cindex[pk];
// check if W[elen + i] empty, skip it
if ((eln = W[elen + i]) <= 0) {
continue;
}
// W[nv + i] was negated
nvi = -W[nv + i];
const wnvi = mark - nvi;
// scan Ei
for (p = cptr[i], p1 = cptr[i] + eln - 1; p <= p1; p++) {
e = cindex[p];
if (W[w + e] >= mark) {
// decrement |Le\Lk|
W[w + e] -= nvi;
} else if (W[w + e] !== 0) {
// ensure e is a live element, 1st time e seen in scan 1
W[w + e] = W[degree + e] + wnvi;
}
}
}
// degree update
// The second pass computes the approximate degree di, prunes the sets Ei and Ai, and computes a hash
// function h(i) for all nodes in Lk.
// scan2: degree update
for (pk = pk1; pk < pk2; pk++) {
// consider node i in Lk
i = cindex[pk];
p1 = cptr[i];
p2 = p1 + W[elen + i] - 1;
pn = p1;
// scan Ei
for (h = 0, d = 0, p = p1; p <= p2; p++) {
e = cindex[p];
// check e is an unabsorbed element
if (W[w + e] !== 0) {
// dext = |Le\Lk|
const dext = W[w + e] - mark;
if (dext > 0) {
// sum up the set differences
d += dext;
// keep e in Ei
cindex[pn++] = e;
// compute the hash of node i
h += e;
} else {
// aggressive absorb. e->k
cptr[e] = (0, _csFlip.csFlip)(k);
// e is a dead element
W[w + e] = 0;
}
}
}
// W[elen + i] = |Ei|
W[elen + i] = pn - p1 + 1;
const p3 = pn;
const p4 = p1 + W[len + i];
// prune edges in Ai
for (p = p2 + 1; p < p4; p++) {
j = cindex[p];
// check node j dead or in Lk
const nvj = W[nv + j];
if (nvj <= 0) {
continue;
}
// degree(i) += |j|
d += nvj;
// place j in node list of i
cindex[pn++] = j;
// compute hash for node i
h += j;
}
// check for mass elimination
if (d === 0) {
// absorb i into k
cptr[i] = (0, _csFlip.csFlip)(k);
nvi = -W[nv + i];
// |Lk| -= |i|
dk -= nvi;
// |k| += W[nv + i]
nvk += nvi;
nel += nvi;
W[nv + i] = 0;
// node i is dead
W[elen + i] = -1;
} else {
// update degree(i)
W[degree + i] = Math.min(W[degree + i], d);
// move first node to end
cindex[pn] = cindex[p3];
// move 1st el. to end of Ei
cindex[p3] = cindex[p1];
// add k as 1st element in of Ei
cindex[p1] = k;
// new len of adj. list of node i
W[len + i] = pn - p1 + 1;
// finalize hash of i
h = (h < 0 ? -h : h) % n;
// place i in hash bucket
W[next + i] = W[hhead + h];
W[hhead + h] = i;
// save hash of i in last[i]
last[i] = h;
}
}
// finalize |Lk|
W[degree + k] = dk;
lemax = Math.max(lemax, dk);
// clear w
mark = _wclear(mark + lemax, lemax, W, w, n);
// Supernode detection. Supernode detection relies on the hash function h(i) computed for each node i.
// If two nodes have identical adjacency lists, their hash functions wil be identical.
for (pk = pk1; pk < pk2; pk++) {
i = cindex[pk];
// check i is dead, skip it
if (W[nv + i] >= 0) {
continue;
}
// scan hash bucket of node i
h = last[i];
i = W[hhead + h];
// hash bucket will be empty
W[hhead + h] = -1;
for (; i !== -1 && W[next + i] !== -1; i = W[next + i], mark++) {
ln = W[len + i];
eln = W[elen + i];
for (p = cptr[i] + 1; p <= cptr[i] + ln - 1; p++) {
W[w + cindex[p]] = mark;
}
let jlast = i;
// compare i with all j
for (j = W[next + i]; j !== -1;) {
let ok = W[len + j] === ln && W[elen + j] === eln;
for (p = cptr[j] + 1; ok && p <= cptr[j] + ln - 1; p++) {
// compare i and j
if (W[w + cindex[p]] !== mark) {
ok = 0;
}
}
// check i and j are identical
if (ok) {
// absorb j into i
cptr[j] = (0, _csFlip.csFlip)(i);
W[nv + i] += W[nv + j];
W[nv + j] = 0;
// node j is dead
W[elen + j] = -1;
// delete j from hash bucket
j = W[next + j];
W[next + jlast] = j;
} else {
// j and i are different
jlast = j;
j = W[next + j];
}
}
}
}
// Finalize new element. The elimination of node k is nearly complete. All nodes i in Lk are scanned one last time.
// Node i is removed from Lk if it is dead. The flagged status of nv[i] is cleared.
for (p = pk1, pk = pk1; pk < pk2; pk++) {
i = cindex[pk];
// check i is dead, skip it
if ((nvi = -W[nv + i]) <= 0) {
continue;
}
// restore W[nv + i]
W[nv + i] = nvi;
// compute external degree(i)
d = W[degree + i] + dk - nvi;
d = Math.min(d, n - nel - nvi);
if (W[head + d] !== -1) {
last[W[head + d]] = i;
}
// put i back in degree list
W[next + i] = W[head + d];
last[i] = -1;
W[head + d] = i;
// find new minimum degree
mindeg = Math.min(mindeg, d);
W[degree + i] = d;
// place i in Lk
cindex[p++] = i;
}
// # nodes absorbed into k
W[nv + k] = nvk;
// length of adj list of element k
if ((W[len + k] = p - pk1) === 0) {
// k is a root of the tree
cptr[k] = -1;
// k is now a dead element
W[w + k] = 0;
}
if (elenk !== 0) {
// free unused space in Lk
cnz = p;
}
}
// Postordering. The elimination is complete, but no permutation has been computed. All that is left
// of the graph is the assembly tree (ptr) and a set of dead nodes and elements (i is a dead node if
// nv[i] is zero and a dead element if nv[i] > 0). It is from this information only that the final permutation
// is computed. The tree is restored by unflipping all of ptr.
// fix assembly tree
for (i = 0; i < n; i++) {
cptr[i] = (0, _csFlip.csFlip)(cptr[i]);
}
for (j = 0; j <= n; j++) {
W[head + j] = -1;
}
// place unordered nodes in lists
for (j = n; j >= 0; j--) {
// skip if j is an element
if (W[nv + j] > 0) {
continue;
}
// place j in list of its parent
W[next + j] = W[head + cptr[j]];
W[head + cptr[j]] = j;
}
// place elements in lists
for (e = n; e >= 0; e--) {
// skip unless e is an element
if (W[nv + e] <= 0) {
continue;
}
if (cptr[e] !== -1) {
// place e in list of its parent
W[next + e] = W[head + cptr[e]];
W[head + cptr[e]] = e;
}
}
// postorder the assembly tree
for (k = 0, i = 0; i <= n; i++) {
if (cptr[i] === -1) {
k = (0, _csTdfs.csTdfs)(i, k, W, head, next, P, w);
}
}
// remove last item in array
P.splice(P.length - 1, 1);
// return P
return P;
};
/**
* Creates the matrix that will be used by the approximate minimum degree ordering algorithm. The function accepts the matrix M as input and returns a permutation
* vector P. The amd algorithm operates on a symmetrix matrix, so one of three symmetric matrices is formed.
*
* Order: 0
* A natural ordering P=null matrix is returned.
*
* Order: 1
* Matrix must be square. This is appropriate for a Cholesky or LU factorization.
* P = M + M'
*
* Order: 2
* Dense columns from M' are dropped, M recreated from M'. This is appropriatefor LU factorization of unsymmetric matrices.
* P = M' * M
*
* Order: 3
* This is best used for QR factorization or LU factorization is matrix M has no dense rows. A dense row is a row with more than 10*sqr(columns) entries.
* P = M' * M
*/
function _createTargetMatrix(order, a, m, n, dense) {
// compute A'
const at = transpose(a);
// check order = 1, matrix must be square
if (order === 1 && n === m) {
// C = A + A'
return add(a, at);
}
// check order = 2, drop dense columns from M'
if (order === 2) {
// transpose arrays
const tindex = at._index;
const tptr = at._ptr;
// new column index
let p2 = 0;
// loop A' columns (rows)
for (let j = 0; j < m; j++) {
// column j of AT starts here
let p = tptr[j];
// new column j starts here
tptr[j] = p2;
// skip dense col j
if (tptr[j + 1] - p > dense) {
continue;
}
// map rows in column j of A
for (const p1 = tptr[j + 1]; p < p1; p++) {
tindex[p2++] = tindex[p];
}
}
// finalize AT
tptr[m] = p2;
// recreate A from new transpose matrix
a = transpose(at);
// use A' * A
return multiply(at, a);
}
// use A' * A, square or rectangular matrix
return multiply(at, a);
}
/**
* Initialize quotient graph. There are four kind of nodes and elements that must be represented:
*
* - A live node is a node i (or a supernode) that has not been selected as a pivot nad has not been merged into another supernode.
* - A dead node i is one that has been removed from the graph, having been absorved into r = flip(ptr[i]).
* - A live element e is one that is in the graph, having been formed when node e was selected as the pivot.
* - A dead element e is one that has benn absorved into a subsequent element s = flip(ptr[e]).
*/
function _initializeQuotientGraph(n, cptr, W, len, head, last, next, hhead, nv, w, elen, degree) {
// Initialize quotient graph
for (let k = 0; k < n; k++) {
W[len + k] = cptr[k + 1] - cptr[k];
}
W[len + n] = 0;
// initialize workspace
for (let i = 0; i <= n; i++) {
// degree list i is empty
W[head + i] = -1;
last[i] = -1;
W[next + i] = -1;
// hash list i is empty
W[hhead + i] = -1;
// node i is just one node
W[nv + i] = 1;
// node i is alive
W[w + i] = 1;
// Ek of node i is empty
W[elen + i] = 0;
// degree of node i
W[degree + i] = W[len + i];
}
// clear w
const mark = _wclear(0, 0, W, w, n);
// n is a dead element
W[elen + n] = -2;
// n is a root of assembly tree
cptr[n] = -1;
// n is a dead element
W[w + n] = 0;
// return mark
return mark;
}
/**
* Initialize degree lists. Each node is placed in its degree lists. Nodes of zero degree are eliminated immediately. Nodes with
* degree >= dense are alsol eliminated and merged into a placeholder node n, a dead element. Thes nodes will appera last in the
* output permutation p.
*/
function _initializeDegreeLists(n, cptr, W, degree, elen, w, dense, nv, head, last, next) {
// result
let nel = 0;
// loop columns
for (let i = 0; i < n; i++) {
// degree @ i
const d = W[degree + i];
// check node i is empty
if (d === 0) {
// element i is dead
W[elen + i] = -2;
nel++;
// i is a root of assembly tree
cptr[i] = -1;
W[w + i] = 0;
} else if (d > dense) {
// absorb i into element n
W[nv + i] = 0;
// node i is dead
W[elen + i] = -1;
nel++;
cptr[i] = (0, _csFlip.csFlip)(n);
W[nv + n]++;
} else {
const h = W[head + d];
if (h !== -1) {
last[h] = i;
}
// put node i in degree list d
W[next + i] = W[head + d];
W[head + d] = i;
}
}
return nel;
}
function _wclear(mark, lemax, W, w, n) {
if (mark < 2 || mark + lemax < 0) {
for (let k = 0; k < n; k++) {
if (W[w + k] !== 0) {
W[w + k] = 1;
}
}
mark = 2;
}
// at this point, W [0..n-1] < mark holds
return mark;
}
function _diag(i, j) {
return i !== j;
}
});