# Enhancing the Performance of Graph Algorithms

We use various types of Data Structures according to the need we have. There are numerous Data Structures available in Computer Science and we are going to see about a brilliant one among them, Heaps! ## So, what is Graph? 1 — A complex graph ...

We use various types of Data Structures according to the need we have. There are numerous Data Structures available in Computer Science and we are going to see about a brilliant one among them, Heaps!

## So, what is Graph?

1 — A complex graph

Graphs are widely used nowadays. They are used in economics, aviation, physics, biology (for DNA analysis), mathematics, and other fields.

2 — Simple graph with notations

A graph is a non-linear Data Structure with nodes(vertices) and edges in Computer Science that is used to implement the undirected graph as well as directed graph theories from the domain of graph theory within Mathematics.

3 — Types of graphs

One of the crucial operations that can be performed on graphs is traversing or searching. There are several algorithms that work on the Graphs.

1. A star algorithm
2. Dijkstra’s shortest path algorithm
3. Bellman-Ford algorithm
4. Floyd cycle detection algorithm
5. Brent’s algorithm
6. Prim’s algorithm
7. Kruskal’s algorithm
8. Kosaraju’s algorithm
9. Tarjan’s strongly connected components algorithm
10. Kahn’s algorithm
11. The algorithm based on a depth-first search
12. Algorithms using the breadth-first search or depth-first search
13. Greedy coloring
14. Ford-Fulkerson algorithm
15. Edmonds–Karp algorithm
16. Dinic’s algorithm
17. Hopcroft-Karp algorithm
18. Hungarian algorithm
19. Blossom algorithm

## So, what is a heap?

A heap is a type of tree-based data structure that fulfills the heap property.

Heap Property: If Q is a child node of P, then key(P) ≥ key (Q). As a result, an element with the greatest key is always in the root node, and such a heap is sometimes referred to as a max-heap.

4 — Two types of heap

There is, of course, a min-heap. For a min-heap, the parent node will always lesser than the child nodes. The root node will have the lowest element.

5 — Same elements in two different heaps

## Let’s see how a min-heap is working

As in the definition, the min-heap has to follow the heap property. That is, if Q is a child node of P, then key(P) ≤ key (Q).

6 — Min-heap construction

The heap can also be stored in an array, which is a more common approach. Because heap is always a complete binary tree, it can be stored in a small amount of space. Pointers take up no space; instead, the parent and children of each node can be found using simple arithmetic on array indices.

7 — Heap in an array

The rules (assume the root is stored in arr[0]). For each index i, element arr[i] has children at arr[2_i_ + 1] and arr[2_i_ + 2], and the parent at arr[floor( ( i — 1 )/2 )].

This implementation is especially useful in the heapsort algorithm, as it allows the space in the input array to be reused for heap storage (i.e., the algorithm is in place). However, it necessitates allocating the array before filling it, making this method unsuitable for implementing priority queues where the number of elements is unknown.

It is completely acceptable to implement a binary heap using a traditional binary tree data structure. When adding an element, there is a problem finding the adjacent element on the last level of the binary heap.

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