Graph Greedy Algorithms
CS 240 Fall 1992 Craig A. Rich
Single-source least-cost paths problem Dijkstras algorithm nds least-cost paths from vertex SOURCE to every vertex in the digraph represented by the labeled adjacency matrix A. LEAST(j) is the cost of a least-cost path from SOURCE to j, or INFINITY if no path exists. PARENT(j) is the parent of j in the tree of least-cost paths from SOURCE, or 0 if j is the root.
procedure DIJKSTRA (A: SOURCE: LEAST: PARENT:
COST_MATRIX; VERTEX; out COST_VECTOR; out COST_VECTOR) is
L: COST_VECTOR := (others => INFINITY); CERTAIN: array (VERTEX) of BOOLEAN := (others => FALSE); function CLOSEST_UNCERTAIN return VERTEX is CLOSEST: NATURAL := INFINITY; k: VERTEX; begin for j in VERTEX loop if not CERTAIN(j) and then L(j) < CLOSEST then CLOSEST := L(j); k := j; end if; end loop; return k; end CLOSEST_UNCERTAIN; procedure MAKE_CERTAIN (k: VERTEX) is begin CERTAIN(k) := TRUE; for j in VERTEX loop if not CERTAIN(j) and then L(k) + A(k,j) < L(j) then L(j) := L(k) + A(k,j); PARENT(j) := NATURAL(k); end if; end loop; end MAKE_CERTAIN; begin PARENT := (others => 0); L(SOURCE) := 0; for k in VERTEX loop MAKE_CERTAIN (CLOSEST_UNCERTAIN); end loop; LEAST := L; end DIJKSTRA;
Minimum-cost tree spanning problem Prims algorithm nds a minimum-cost spanning tree of the connected graph represented by the symmetric labeled adjacency matrix A. PARENT(j) is the parent of j in the minimum-cost spanning tree, or 0 if j is the root. LEAST(j) is the cost of the edge {j, PARENT(j)} in the minimum-cost spanning tree, or 0 if j is the root; therefore, the minimum-cost spanning tree has cost
n
LEAST(j).
j=1
procedure PRIM
(A: SOURCE: LEAST: PARENT:
COST_MATRIX; VERTEX; out COST_VECTOR; out COST_VECTOR) is
L: COST_VECTOR := (others => INFINITY); CERTAIN: array (VERTEX) of BOOLEAN := (others => FALSE); function CLOSEST_UNCERTAIN return VERTEX is CLOSEST: NATURAL := INFINITY; k: VERTEX; begin for ...