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Joined 2026-09-21 Message
Incoming links: Indexed priority queue
d-ary min-heap 2026-10-31
A d-ary heap in which each parent key is no greater than each child key. Transitivity makes the root a minimum. Removing the root requires choosing a smallest child at each downward step, taking time in the usual range. Decreasing a key or inserting a new leaf only compares along one chain of parents, taking time in that range. These priority queue bounds assume constant-time key comparisons and available array capacity. An indexed priority queue must also update its item-to-index table after every swap.
Use an indexed priority queue of unsettled vertices, keyed by their tentative distances. Initialize the source to zero and other keys to . Each
ExtractMin settles the currently closest vertex. Scan its outgoing edges; whenever edge relaxation improves a neighbour's tentative distance, update that neighbour with DecreaseKey. Already settled vertices need no further update, by the correctness of Dijkstra's algorithm for nonnegative weights.There are at most extractions and at most successful decreases. The indexed priority queue maintains each vertex's current array position and updates this index table on every swap, so locating a key for
DecreaseKey costs . Preallocate capacity . Repeated insertion costs , or bottom-up heap construction initializes the queue in time. Scanning adjacency lists costs in total.For and , the heap-operation contributions are thereforeand the overall bound isA uniform bound including larger arity and tiny graphs is . Arity greater than the number of vertices offers no useful extra heap levels. These bounds assume adjacency lists; with an adjacency matrix, scanning neighbours contributes instead. No rebuilding of the heap is required after each decrease.
Prim's algorithm grows a tree from any root . For each vertex outside the current tree, its priority queue key is the cheapest single edge connecting it to the tree, not the total length of a root-to-vertex path:The indexed priority queue supports membership and locating a vertex's heap entry. After the first extraction, the chosen parent edge is a least-weight edge crossing the cut between the current tree and its complement. The minimum spanning tree cut property makes that edge safe: exchanging it for a crossing edge of a suitable minimum tree preserves minimality. Repeating this establishes a minimum tree. Connectivity ensures every nonroot vertex eventually has a parent; edge weights may be negative.
for v in V:
key[v] = infinity
parent[v] = NONE
key[r] = 0
Q = indexed_min_priority_queue(V, key)
while Q is not empty:
u = Q.extract_min()
for (v,weight) in adjacency[u]:
if Q.contains(v) and weight < key[v]:
parent[v] = u
key[v] = weight
Q.decrease_key(v, weight)
return {(parent[v],v) : v != r}