cs_amd.c

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#include "cs.h"
/* clear w */
static int cs_wclear (int mark, int lemax, int *w, int n)
{
    int k ;
    if (mark < 2 || (mark + lemax < 0))
    {
        for (k = 0 ; k < n ; k++) if (w [k] != 0) w [k] = 1 ;
        mark = 2 ;
    }
    return (mark) ;     /* at this point, w [0..n-1] < mark holds */
}

/* keep off-diagonal entries; drop diagonal entries */
static int cs_diag (int i, int j, double aij, void *other) { return (i != j) ; }

/* p = amd(A+A') if symmetric is true, or amd(A'A) otherwise */
int *cs_amd (int order, const cs *A)  /* order 0:natural, 1:Chol, 2:LU, 3:QR */
{
    cs *C, *A2, *AT ;
    int *Cp, *Ci, *last, *W, *len, *nv, *next, *P, *head, *elen, *degree, *w,
        *hhead, *ATp, *ATi, d, dk, dext, lemax = 0, e, elenk, eln, i, j, k, k1,
        k2, k3, jlast, ln, dense, nzmax, mindeg = 0, nvi, nvj, nvk, mark, wnvi,
        ok, cnz, nel = 0, p, p1, p2, p3, p4, pj, pk, pk1, pk2, pn, q, n, m, t ;
    unsigned int h ;
    /* --- Construct matrix C ----------------------------------------------- */
    if (!CS_CSC (A) || order <= 0 || order > 3) return (NULL) ; /* check */
    AT = cs_transpose (A, 0) ;              /* compute A' */
    if (!AT) return (NULL) ;
    m = A->m ; n = A->n ;
    dense = CS_MAX (16, 10 * sqrt ((double) n)) ;   /* find dense threshold */
    dense = CS_MIN (n-2, dense) ;
    if (order == 1 && n == m)
    {
        C = cs_add (A, AT, 0, 0) ;          /* C = A+A' */
    }
    else if (order == 2)
    {
        ATp = AT->p ;                       /* drop dense columns from AT */
        ATi = AT->i ;
        for (p2 = 0, j = 0 ; j < m ; j++)
        {
            p = ATp [j] ;                   /* column j of AT starts here */
            ATp [j] = p2 ;                  /* new column j starts here */
            if (ATp [j+1] - p > dense) continue ;   /* skip dense col j */
            for ( ; p < ATp [j+1] ; p++) ATi [p2++] = ATi [p] ;
        }
        ATp [m] = p2 ;                      /* finalize AT */
        A2 = cs_transpose (AT, 0) ;         /* A2 = AT' */
        C = A2 ? cs_multiply (AT, A2) : NULL ;  /* C=A'*A with no dense rows */
        cs_spfree (A2) ;
    }
    else
    {
        C = cs_multiply (AT, A) ;           /* C=A'*A */
    }
    cs_spfree (AT) ;
    if (!C) return (NULL) ;
    cs_fkeep (C, &cs_diag, NULL) ;          /* drop diagonal entries */
    Cp = C->p ;
    cnz = Cp [n] ;
    P = cs_malloc (n+1, sizeof (int)) ;     /* allocate result */
    W = cs_malloc (8*(n+1), sizeof (int)) ; /* get workspace */
    t = cnz + cnz/5 + 2*n ;                 /* add elbow room to C */
    if (!P || !W || !cs_sprealloc (C, t)) return (cs_idone (P, C, W, 0)) ;
    len  = W           ; nv     = W +   (n+1) ; next   = W + 2*(n+1) ;
    head = W + 3*(n+1) ; elen   = W + 4*(n+1) ; degree = W + 5*(n+1) ;
    w    = W + 6*(n+1) ; hhead  = W + 7*(n+1) ;
    last = P ;                              /* use P as workspace for last */
    /* --- Initialize quotient graph ---------------------------------------- */
    for (k = 0 ; k < n ; k++) len [k] = Cp [k+1] - Cp [k] ;
    len [n] = 0 ;
    nzmax = C->nzmax ;
    Ci = C->i ;
    for (i = 0 ; i <= n ; i++)
    {
        head [i] = -1 ;                     /* degree list i is empty */
        last [i] = -1 ;
        next [i] = -1 ;
        hhead [i] = -1 ;                    /* hash list i is empty */
        nv [i] = 1 ;                        /* node i is just one node */
        w [i] = 1 ;                         /* node i is alive */
        elen [i] = 0 ;                      /* Ek of node i is empty */
        degree [i] = len [i] ;              /* degree of node i */
    }
    mark = cs_wclear (0, 0, w, n) ;         /* clear w */
    elen [n] = -2 ;                         /* n is a dead element */
    Cp [n] = -1 ;                           /* n is a root of assembly tree */
    w [n] = 0 ;                             /* n is a dead element */
    /* --- Initialize degree lists ------------------------------------------ */
    for (i = 0 ; i < n ; i++)
    {
        d = degree [i] ;
        if (d == 0)                         /* node i is empty */
        {
            elen [i] = -2 ;                 /* element i is dead */
            nel++ ;
            Cp [i] = -1 ;                   /* i is a root of assembly tree */
            w [i] = 0 ;
        }
        else if (d > dense)                 /* node i is dense */
        {
            nv [i] = 0 ;                    /* absorb i into element n */
            elen [i] = -1 ;                 /* node i is dead */
            nel++ ;
            Cp [i] = CS_FLIP (n) ;
            nv [n]++ ;
        }
        else
        {
            if (head [d] != -1) last [head [d]] = i ;
            next [i] = head [d] ;           /* put node i in degree list d */
            head [d] = i ;
        }
    }
    while (nel < n)                         /* while (selecting pivots) do */
    {
        /* --- Select node of minimum approximate degree -------------------- */
        for (k = -1 ; mindeg < n && (k = head [mindeg]) == -1 ; mindeg++) ;
        if (next [k] != -1) last [next [k]] = -1 ;
        head [mindeg] = next [k] ;          /* remove k from degree list */
        elenk = elen [k] ;                  /* elenk = |Ek| */
        nvk = nv [k] ;                      /* # of nodes k represents */
        nel += nvk ;                        /* nv[k] nodes of A eliminated */
        /* --- Garbage collection ------------------------------------------- */
        if (elenk > 0 && cnz + mindeg >= nzmax)
        {
            for (j = 0 ; j < n ; j++)
            {
                if ((p = Cp [j]) >= 0)      /* j is a live node or element */
                {
                    Cp [j] = Ci [p] ;       /* save first entry of object */
                    Ci [p] = CS_FLIP (j) ;  /* first entry is now CS_FLIP(j) */
                }
            }
            for (q = 0, p = 0 ; p < cnz ; ) /* scan all of memory */
            {
                if ((j = CS_FLIP (Ci [p++])) >= 0)  /* found object j */
                {
                    Ci [q] = Cp [j] ;       /* restore first entry of object */
                    Cp [j] = q++ ;          /* new pointer to object j */
                    for (k3 = 0 ; k3 < len [j]-1 ; k3++) Ci [q++] = Ci [p++] ;
                }
            }
            cnz = q ;                       /* Ci [cnz...nzmax-1] now free */
        }
        /* --- Construct new element ---------------------------------------- */
        dk = 0 ;
        nv [k] = -nvk ;                     /* flag k as in Lk */
        p = Cp [k] ;
        pk1 = (elenk == 0) ? p : cnz ;      /* do in place if elen[k] == 0 */
        pk2 = pk1 ;
        for (k1 = 1 ; k1 <= elenk + 1 ; k1++)
        {
            if (k1 > elenk)
            {
                e = k ;                     /* search the nodes in k */
                pj = p ;                    /* list of nodes starts at Ci[pj]*/
                ln = len [k] - elenk ;      /* length of list of nodes in k */
            }
            else
            {
                e = Ci [p++] ;              /* search the nodes in e */
                pj = Cp [e] ;
                ln = len [e] ;              /* length of list of nodes in e */
            }
            for (k2 = 1 ; k2 <= ln ; k2++)
            {
                i = Ci [pj++] ;
                if ((nvi = nv [i]) <= 0) continue ; /* node i dead, or seen */
                dk += nvi ;                 /* degree[Lk] += size of node i */
                nv [i] = -nvi ;             /* negate nv[i] to denote i in Lk*/
                Ci [pk2++] = i ;            /* place i in Lk */
                if (next [i] != -1) last [next [i]] = last [i] ;
                if (last [i] != -1)         /* remove i from degree list */
                {
                    next [last [i]] = next [i] ;
                }
                else
                {
                    head [degree [i]] = next [i] ;
                }
            }
            if (e != k)
            {
                Cp [e] = CS_FLIP (k) ;      /* absorb e into k */
                w [e] = 0 ;                 /* e is now a dead element */
            }
        }
        if (elenk != 0) cnz = pk2 ;         /* Ci [cnz...nzmax] is free */
        degree [k] = dk ;                   /* external degree of k - |Lk\i| */
        Cp [k] = pk1 ;                      /* element k is in Ci[pk1..pk2-1] */
        len [k] = pk2 - pk1 ;
        elen [k] = -2 ;                     /* k is now an element */
        /* --- Find set differences ----------------------------------------- */
        mark = cs_wclear (mark, lemax, w, n) ;  /* clear w if necessary */
        for (pk = pk1 ; pk < pk2 ; pk++)    /* scan 1: find |Le\Lk| */
        {
            i = Ci [pk] ;
            if ((eln = elen [i]) <= 0) continue ;/* skip if elen[i] empty */
            nvi = -nv [i] ;                      /* nv [i] was negated */
            wnvi = mark - nvi ;
            for (p = Cp [i] ; p <= Cp [i] + eln - 1 ; p++)  /* scan Ei */
            {
                e = Ci [p] ;
                if (w [e] >= mark)
                {
                    w [e] -= nvi ;          /* decrement |Le\Lk| */
                }
                else if (w [e] != 0)        /* ensure e is a live element */
                {
                    w [e] = degree [e] + wnvi ; /* 1st time e seen in scan 1 */
                }
            }
        }
        /* --- Degree update ------------------------------------------------ */
        for (pk = pk1 ; pk < pk2 ; pk++)    /* scan2: degree update */
        {
            i = Ci [pk] ;                   /* consider node i in Lk */
            p1 = Cp [i] ;
            p2 = p1 + elen [i] - 1 ;
            pn = p1 ;
            for (h = 0, d = 0, p = p1 ; p <= p2 ; p++)    /* scan Ei */
            {
                e = Ci [p] ;
                if (w [e] != 0)             /* e is an unabsorbed element */
                {
                    dext = w [e] - mark ;   /* dext = |Le\Lk| */
                    if (dext > 0)
                    {
                        d += dext ;         /* sum up the set differences */
                        Ci [pn++] = e ;     /* keep e in Ei */
                        h += e ;            /* compute the hash of node i */
                    }
                    else
                    {
                        Cp [e] = CS_FLIP (k) ;  /* aggressive absorb. e->k */
                        w [e] = 0 ;             /* e is a dead element */
                    }
                }
            }
            elen [i] = pn - p1 + 1 ;        /* elen[i] = |Ei| */
            p3 = pn ;
            p4 = p1 + len [i] ;
            for (p = p2 + 1 ; p < p4 ; p++) /* prune edges in Ai */
            {
                j = Ci [p] ;
                if ((nvj = nv [j]) <= 0) continue ; /* node j dead or in Lk */
                d += nvj ;                  /* degree(i) += |j| */
                Ci [pn++] = j ;             /* place j in node list of i */
                h += j ;                    /* compute hash for node i */
            }
            if (d == 0)                     /* check for mass elimination */
            {
                Cp [i] = CS_FLIP (k) ;      /* absorb i into k */
                nvi = -nv [i] ;
                dk -= nvi ;                 /* |Lk| -= |i| */
                nvk += nvi ;                /* |k| += nv[i] */
                nel += nvi ;
                nv [i] = 0 ;
                elen [i] = -1 ;             /* node i is dead */
            }
            else
            {
                degree [i] = CS_MIN (degree [i], d) ;   /* update degree(i) */
                Ci [pn] = Ci [p3] ;         /* move first node to end */
                Ci [p3] = Ci [p1] ;         /* move 1st el. to end of Ei */
                Ci [p1] = k ;               /* add k as 1st element in of Ei */
                len [i] = pn - p1 + 1 ;     /* new len of adj. list of node i */
                h %= n ;                    /* finalize hash of i */
                next [i] = hhead [h] ;      /* place i in hash bucket */
                hhead [h] = i ;
                last [i] = h ;              /* save hash of i in last[i] */
            }
        }                                   /* scan2 is done */
        degree [k] = dk ;                   /* finalize |Lk| */
        lemax = CS_MAX (lemax, dk) ;
        mark = cs_wclear (mark+lemax, lemax, w, n) ;    /* clear w */
        /* --- Supernode detection ------------------------------------------ */
        for (pk = pk1 ; pk < pk2 ; pk++)
        {
            i = Ci [pk] ;
            if (nv [i] >= 0) continue ;         /* skip if i is dead */
            h = last [i] ;                      /* scan hash bucket of node i */
            i = hhead [h] ;
            hhead [h] = -1 ;                    /* hash bucket will be empty */
            for ( ; i != -1 && next [i] != -1 ; i = next [i], mark++)
            {
                ln = len [i] ;
                eln = elen [i] ;
                for (p = Cp [i]+1 ; p <= Cp [i] + ln-1 ; p++) w [Ci [p]] = mark;
                jlast = i ;
                for (j = next [i] ; j != -1 ; ) /* compare i with all j */
                {
                    ok = (len [j] == ln) && (elen [j] == eln) ;
                    for (p = Cp [j] + 1 ; ok && p <= Cp [j] + ln - 1 ; p++)
                    {
                        if (w [Ci [p]] != mark) ok = 0 ;    /* compare i and j*/
                    }
                    if (ok)                     /* i and j are identical */
                    {
                        Cp [j] = CS_FLIP (i) ;  /* absorb j into i */
                        nv [i] += nv [j] ;
                        nv [j] = 0 ;
                        elen [j] = -1 ;         /* node j is dead */
                        j = next [j] ;          /* delete j from hash bucket */
                        next [jlast] = j ;
                    }
                    else
                    {
                        jlast = j ;             /* j and i are different */
                        j = next [j] ;
                    }
                }
            }
        }
        /* --- Finalize new element------------------------------------------ */
        for (p = pk1, pk = pk1 ; pk < pk2 ; pk++)   /* finalize Lk */
        {
            i = Ci [pk] ;
            if ((nvi = -nv [i]) <= 0) continue ;/* skip if i is dead */
            nv [i] = nvi ;                      /* restore nv[i] */
            d = degree [i] + dk - nvi ;         /* compute external degree(i) */
            d = CS_MIN (d, n - nel - nvi) ;
            if (head [d] != -1) last [head [d]] = i ;
            next [i] = head [d] ;               /* put i back in degree list */
            last [i] = -1 ;
            head [d] = i ;
            mindeg = CS_MIN (mindeg, d) ;       /* find new minimum degree */
            degree [i] = d ;
            Ci [p++] = i ;                      /* place i in Lk */
        }
        nv [k] = nvk ;                      /* # nodes absorbed into k */
        if ((len [k] = p-pk1) == 0)         /* length of adj list of element k*/
        {
            Cp [k] = -1 ;                   /* k is a root of the tree */
            w [k] = 0 ;                     /* k is now a dead element */
        }
        if (elenk != 0) cnz = p ;           /* free unused space in Lk */
    }
    /* --- Postordering ----------------------------------------------------- */
    for (i = 0 ; i < n ; i++) Cp [i] = CS_FLIP (Cp [i]) ;/* fix assembly tree */
    for (j = 0 ; j <= n ; j++) head [j] = -1 ;
    for (j = n ; j >= 0 ; j--)              /* place unordered nodes in lists */
    {
        if (nv [j] > 0) continue ;          /* skip if j is an element */
        next [j] = head [Cp [j]] ;          /* place j in list of its parent */
        head [Cp [j]] = j ;
    }
    for (e = n ; e >= 0 ; e--)              /* place elements in lists */
    {
        if (nv [e] <= 0) continue ;         /* skip unless e is an element */
        if (Cp [e] != -1)
        {
            next [e] = head [Cp [e]] ;      /* place e in list of its parent */
            head [Cp [e]] = e ;
        }
    }
    for (k = 0, i = 0 ; i <= n ; i++)       /* postorder the assembly tree */
    {
        if (Cp [i] == -1) k = cs_tdfs (i, k, head, next, P, w) ;
    }
    return (cs_idone (P, C, W, 1)) ;
}