simple-squiggle/node_modules/mathjs/lib/esm/function/algebra/sparse/csAmd.js

import { factory } from '../../../utils/factory.js';
import { csFkeep } from './csFkeep.js';
import { csFlip } from './csFlip.js';
import { csTdfs } from './csTdfs.js';
var name = 'csAmd';
var dependencies = ['add', 'multiply', 'transpose'];
export var createCsAmd = /* #__PURE__ */factory(name, dependencies, _ref => {
  var {
    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.
   *
   * Reference: http://faculty.cse.tamu.edu/davis/publications.html
   *
   * @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


    var asize = a._size; // rows and columns

    var m = asize[0];
    var n = asize[1]; // initialize vars

    var lemax = 0; // dense threshold

    var dense = Math.max(16, 10 * Math.sqrt(n));
    dense = Math.min(n - 2, dense); // create target matrix C

    var cm = _createTargetMatrix(order, a, m, n, dense); // drop diagonal entries


    csFkeep(cm, _diag, null); // C matrix arrays

    var cindex = cm._index;
    var cptr = cm._ptr; // number of nonzero elements in C

    var cnz = cptr[n]; // allocate result (n+1)

    var P = []; // create workspace (8 * (n + 1))

    var W = [];
    var len = 0; // first n + 1 entries

    var nv = n + 1; // next n + 1 entries

    var next = 2 * (n + 1); // next n + 1 entries

    var head = 3 * (n + 1); // next n + 1 entries

    var elen = 4 * (n + 1); // next n + 1 entries

    var degree = 5 * (n + 1); // next n + 1 entries

    var w = 6 * (n + 1); // next n + 1 entries

    var hhead = 7 * (n + 1); // last n + 1 entries
    // use P as workspace for last

    var last = P; // initialize quotient graph

    var mark = _initializeQuotientGraph(n, cptr, W, len, head, last, next, hhead, nv, w, elen, degree); // initialize degree lists


    var nel = _initializeDegreeLists(n, cptr, W, degree, elen, w, dense, nv, head, last, next); // minimum degree node


    var mindeg = 0; // vars

    var 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|

      var elenk = W[elen + k]; // # of nodes k represents

      var 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.

      var dk = 0; // flag k as in Lk

      W[nv + k] = -nvk;
      var p = cptr[k]; // do in place if W[elen + k] === 0

      var pk1 = elenk === 0 ? p : cnz;
      var 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] = 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];
        var 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|
            var 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] = csFlip(k); // e is a dead element

              W[w + e] = 0;
            }
          }
        } // W[elen + i] = |Ei|


        W[elen + i] = pn - p1 + 1;
        var p3 = pn;
        var 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

          var 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] = 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;
          }

          var jlast = i; // compare i with all j

          for (j = W[next + i]; j !== -1;) {
            var 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] = 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] = 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 = 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'
    var 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
      var tindex = at._index;
      var tptr = at._ptr; // new column index

      var p2 = 0; // loop A' columns (rows)

      for (var j = 0; j < m; j++) {
        // column j of AT starts here
        var 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 (var 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 (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;
  }
});