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