Kruskal's algorithm finds a minimum spanning tree by sorting the edges by weight and then iterating through each edge, adding it to the spanning tree if it does not form a cycle. It uses a disjoint-set union find data structure to check for cycles in O(E log V) time, resulting in an overall runtime of O(E log E). The algorithm puts edges in a min-heap based on their weights and iterates through them, adding each edge to the output if its endpoints are in different sets in the disjoint-set union find structure.

1. Kruskal’s algorithm
S.RENUGADEVI,
MSC(CS).,
NADAR SARASWATHI COLLEGE OF ARTS
AND SCIENCE, THENI

2. IDEAS OF KRUSKALS
ALGORITHM
 CENTRAL IDEA:
 Grow a forest out of edges that do not grow a cycle, just like
for the spanning tree problem.
 But now consider the edges in order by weight
 BASIC IMPLEMENTATION:
 Sort edges by weight - >o(E log E) = o(E log E)
 Iterate through edges using DSUF for cycle detection -> o(E
log V)

3. IMPLEMENTATION
 Floyds algorithm to build min-heap with
edges->0(E)
 Iterate through edges using DSUF for cycle
detection and detemin to get next edge->0(E
log V)
 Not better worst-case asymptotically, but
often stop long before considering all edges.

4. PSEUDOCODE:
 Put edges in min-heap using edge weights
 Create DSUF with each vertex in its own set
 While output size <V-1
consider next smallest edge(u,v)
if find (u,v) indicates u and v are in
different sets

5.  Output(u,v)
 Union(u,v)
RECALL INVARIANT:
u and v in same set if and only if
connected in output-so-far.

15. Thanking you…………..