D-ary, Indexed & Mergeable Heaps

Engineering variants: wider nodes (d-ary), position tracking for decrease-key (indexed), and heaps that merge in O(log n) (pairing, Fibonacci).

What to know

D-ary heaps trade deeper sift-downs for shallower trees — 4-ary often wins in practice for Dijkstra.
Indexed priority queues map item → heap position so priorities can be updated in place.
Fibonacci heaps give O(1) amortized decrease-key (theory); pairing heaps win in practice.

In the wild: Network routing daemons, simulation engines merging event queues, game AI planners.

Dijkstra with decrease-key

O(E + V log V) w/ Fib

Update tentative distances in place

Pairing-heap meld

O(1) amortized

Merge two priority queues cheaply

D-ary sift tuning

O(log_d n) per op

Cache-tuned priority queues

Each problem has its own page: progressive hints, how to approach it, and the full solution.

Network Delay Time

Hints → approach → solution

Minimum Cost to Hire K Workers

Hints → approach → solution