Sorting

Bubble Sort

  • Best case Time Complexity: $O(n)$ since n comparison and no swaps
  • Worst case Time Complexity: $O(n^2)$ since n^2 comparisons and n^2 swaps
	public static <E extends Comparable<E>> void bubbleSort(ArrayList<E> arr) {
		for (int i = arr.size() - 1; i >= 0; i--) {
			int swaps = 0;
			for (int j = 0; j < i; j++) {
				if (arr.get(j).compareTo(arr.get(j + 1)) > 0) {
					swap(arr, j, j+1);
					swaps++;
				}
			}
			if (swaps == 0) {
				break;
			}
		}
	}

Selection Sort

  • Best case Time Complexity: $O(n^2)$ since n^2 comparison and n in place swaps (0 swaps)
  • Worst case Time Complexity: $O(n^2)$ since n^2 comparisons and n swaps
	public static <E extends Comparable<E>> void selectionSort(ArrayList<E> arr) {
		for (int i = 0; i < arr.size() - 1; i++) {
			int min_idx = i;
			for (int j = i; j < arr.size(); j++) {
				if (arr.get(j).compareTo(arr.get(min_idx)) < 0) {
					min_idx = j;
				}
			}
			swap(arr, i, min_idx);
			if (i != min_idx) {
			}
		}
	}

Insertion Sort

  • Online: works with bits of the data (a stream); the entire input isn’t needed all at once, the algorithm can work by feeding input gradually

  • Adaptive: Adding new extra data doesn’t mean you have to start the sorting algo over again. Works better with already sorted data.

  • Best case Time Complexity: $O(n)$ since O(n) comparison and 0 swaps

  • Worst case Time Complexity: $O(n^2)$ since around O(n^2) comparisons and O(n^2) swaps

	// Based on Listing 8.2 InsertionSort example from "Data Structures Abstraction and Design Using Java" by Koffman and Wolfgang
	public static <E extends Comparable<E>> void insertionSort(ArrayList<E> arr) {
		// index i keeps track of the element we're inserting rn
		// index i is also the length of the sorted portion of the array
		for (int  i = 1; i < arr.size(); i++) {
			E itemToInsert = arr.get(i);
			// index j iterates over the sorted portion of the array
			// to find where to insert the item at i
			// the for loop also at the same time shifts all the elements of unsorted portion of the arr
			// to the right to make room for the eventual insertion of itemToInsert
			int j;
			for (j = i - 1; j >= 0 && arr.get(j).compareTo(itemToInsert) > 0; j--) {
				arr.set(j + 1, arr.get(j));
			}
			// We've found the position to insert element at i
			// if arr.get(j + 1) > arr.get(i)
			arr.set(j + 1, itemToInsert);
		}
	}

Quicksort

  • Best/Average case Time Complexity: $O(n \log{(n)})$

    • Array is halved each recursive call
    • Hand wavy explanation: Recursive depth is just $O(log(n))$ and partition at each call is just $O(n)$ (more mathy explanation similar to merge sort time complexity)
  • Worst case Time Complexity: $O(n^2)$

    • Occurs when all the elements are on one side of the pivot on every recursive call.
    • Thus all $n$ recursive calls happen in the same branch and thus the depth of the recusion tree is $n$. Each call does a partition which takes an average of $n$ each call.
  • Best/Average Case Space Complexity: $O(log(n))$

    • For an average/best case, the recursion tree is only $log(n)$ calls deep. Even though a total of $n$ recursive calls are made, only $log(n)$ stack frames exist at a given time.
  • Worst Case Space Complexity: $O(n)$

    • In the worst case, $n$ recursive calls are made, but this time instead of splitting the array in half in each recursive call, only one element (the pivot) is split off. Thus the recursion tree is $n$ stack frames deep.

The algorithm works by choosing a pivot point and partitioning the array into elements smaller than the pivot and elements greater/equal to the pivot. Then the algorithm is recursively called on the two halves. The base case for the recursive quickSort would be when the first and last pointers cross (the pointers keep track of the bounds of the current input for a call).

Pseudocode for (unstable) partition algorithm:

// Pseudocode from ch08.pptx slide 370 
// https://youtu.be/1nCog_qwCdU?list=PLNDWoTOY5hTaVWg1Ar8ztvaH7qwq_bgcw&t=506
// or pg 413 of "Data Structures Abstraction and Design Using Java" by Koffman and Wolfgang

// Note in this implementation, the pivot is first at the beginning and then moved where it belongs in the array (best case in the middle of the array)

1. Choose pivot
	check if first is median
	check if mid is median
	else last is the median

	Mov pivot to front
2. Initialize *up* to *first* and *down* to *last*
3. do
	4. Increment up until up selects the first element greater than the pivot or has reached last
	5. Decrement down until down selects the first element less than or equal to the pivot value or down has reached first.
	6. if up < down then 
		7. Exchange table[up] and table[down]
8. while up is to the left of down (up < down)
9. Exchange table[first] and table[down] (exchange pivot with elem at down)
10. Return the value of down to pivIndex

First is the index of the first element of the current input
Last is the index of the last element of the current input
Up and down are pointers

  • The exchange at line 9 moves the pivot to the right position in the array.
    In this scenario where we originally have the pivot at the beginning, the down pointer is guaranteed to point at an element less than the pivot (by the definition of the down pointer). Thus swapping the pivot with the down pointer’s element will yield the proper ordering where the pivot is in between everything less than the pivot and everything greater/equal to.

Merge Sort

  • Time Complexity: $O(n \log{(n)})$

    • Array is halved each recursive call
    • Hand wavy explanation: Recursive depth is just $O(log(n))$ (always) and merging at each call is just $O(n)$ (more mathy explanation merge sort time complexity)
    • Since recursive depth is always log(n), the best, worst, and average are always $O(log(n))$
  • Space Complexity: $O(n)$

    • Merging is not done in place for most of the time (while in place merging exists, in reality, it makes things slower).
    • The normal version of merge sort uses auxiliary space that is bounded by 2n or just $O(n)$ at any given time in a merge sort call. The most space is used when you’re down in the leaf of the recursion tree at maximal depth. Also note that only one branch is expanded out at a time in the recursion tree.

    $$\sum_{i=1}^{log_2(n) } \frac{2n}{2^i} = 2(n-2) = O(n)$$

    • The sum represents that there is a depth at most of $log_2(n)$ levels, where each level has $n/(2^i)$ auxillary space taken up, where $n$ is the total, initial size of the array input on the first call of merge sort. The first call eventually merges two temporary arrays of size n/2, the second call merges two temp arrays of n/4, then two n/8 arrays, and so on, until the base case is reached (size of 1).
  • Merge algorithm

// Taken from Skiena's Algorithm Design Manual 3rd Edition pg 129-130

void merge(item_type s[], int low, int middle, int high) {
	int i; /* counter */
	queue buffer1, buffer2; /* buffers to hold elements for merging */

	init_queue(&buffer1);
	init_queue(&buffer2);

	for (i = low; i <= middle; i++) enqueue(&buffer1, s[i]);
	for (i = middle + 1; i <= high; i++) enqueue(&buffer2, s[i]);

	i = low;
	while (!(empty_queue(&buffer1) || empty_queue(&buffer2))) {
		if (headq(&buffer1) <= headq(&buffer2)) {
			s[i++] = dequeue(&buffer1);
		} else {
			s[i++] = dequeue(&buffer2);
		}
	}

	while (!empty_queue(&buffer1)) {
		s[i++] = dequeue(&buffer1);
	}
	while (!empty_queue(&buffer2)) {
		s[i++] = dequeue(&buffer2);
	}
}

Heaps

Bubble Up

  • Used only when inserting a new item into an already heapified structure

Bubble Down

  • Used both when removing from a heap and when doing a fast array to heap conversion in $O(n)$ time
  • Bubble_down basically operates when you have a root which is out of place and children of root that have the heap property (subheaps)
  • bubble_down takes $O(log(n))$, where n is the number of nodes in the subtree with the root as the root

Array to Heapified Array

  1. The last n/2 elements are leaves, so start at the back of the array at index $\frac{n}{2}$
  2. Call bubble_down on the element and then decrement the index
  3. Repeat step 3 until you reach the first element

Derivation of O(n) time for array to heapified array

  • Bubble_down is more efficient/faster than calling bubble_up on every element.
  • Even though bubble_down takes $log(n)$, the value of $n$ changes every time you call bubble_down, where $n$ is the total size of the array we’re trying to heapify. There are $\frac{n}{2}$ leaves with subheap of height 0, $\frac{n}{4}$ nodes with subheap of height 1, $\frac{n}{8}$ nodes with subheap of height 2, and so on. In general, there are $n/2^{h+1}$ nodes of height $h$.

$$ \sum_{h=0}^{\lg n} n/2^{h+1}\cdot h \le n \sum_{h=0}^{\lg n} h/2^h \le 2n$$

  • You can use the Taylor Series to derive the above.
  • Above formula taken from the Algorithm Design Manual

From the stackoverflow post:

The number of operations required for siftDown and siftUp is proportional to the distance the node may have to move. For siftDown, it is the distance to the bottom of the tree, so siftDown is expensive for nodes at the top of the tree. With siftUp, the work is proportional to the distance to the top of the tree, so siftUp is expensive for nodes at the bottom of the tree. Although both operations are O(log n) in the worst case, in a heap, only one node is at the top whereas half the nodes lie in the bottom layer. So it shouldn’t be too surprising that if we have to apply an operation to every node, we would prefer siftDown over siftUp.

Graphs

Terminology

  • Simple: No multiple edges, no loops, unweighted, and undirected
  • Connected: All nodes have a path/edge to at least one other node/vertex
  • Acyclic: No cycles
  • Directed: Edges have directions

Common Algorithms

  • Notes based on slides from the “Data Structures Abstraction and Design Using Java” book

Some Graph Algorithms are of the form:

1. for each vertex u in the graph:
2.   for each vertex v adjacent to u
3.     do something with edge(u, v)
  • For adjacency list

    • Step 1 is $O(|V|)$
    • Step 2 is $O(|E_U|)$
    • Combination of Steps 1 and 2 represents examining each edge in the graph, giving $O(|E|)$ $$\sum_{u=1}^{|V|} \left(\sum_{v=1}^{|E_u|} O(1)\right) = |E|$$
  • For adjacency matrices

    • Step 1 is $O(|V|)$
    • Step 2 is $O(|V|)$
    • Combination of Steps 1 and 2 give $O(|V|^2)$ $$\sum_{u=1}^{|V|} \left(\sum_{v=1}^{|V|} O(1)\right) = |V|^2$$

Other graphs algorithms are of the form:

1. for each vertex u in some subset of the vertices
2.   for each vertex v in some subset of the vertices
3.     if (u, v) is an edge
4.       do something with edge(u, v)
  • Adjacency List:

    • Step 3 searches a list and is $O(|E_v|)$
    • Step 2 and 3 take $O(|E|)$
    • The overall algo is $O(|V||E|)$ $$\sum_{u=1}^{|V|} \left(\sum_{v=1}^{|V|} \left(\sum_{n=1}^{|E_v|} O(1)\right)\right) = \sum_{u=1}^{|V|}|E| = |V|\cdot|E|$$
  • Adjacency Matrix:

    • Step 3 is just $O(1)$
    • Overall is just $O(|V|^2)$ $$\sum_{u=1}^{|V|} \left(\sum_{v=1}^{|V|} O(1)\right) = \sum_{u=1}^{|V|}|V| = |V|^2$$
  • An adjacency matrix is better for dense graphs

  • Adjacency list is better for sparse graphs

  • Break-even point: If adjacency matrix is filled 25%

  • Identified = put in queue and marked as known to exist, but not yet processed/visited
  • Visited = Done processing
  • Unknown = Not yet known to exist in graph
  • Can be used to find the shortest path for unweighted edges
1. Take arbitrary start vertex and mark as identified and put in queue
2. while the queue is not empty
3.   take a vertex, u, out of the queue and visit u
4.   for all vertices, v, adjacent to the vertex u
5.     if v has not been identified or visited
6.       mark it identified
7.       insert v into the queue
8.   we are now finished visiting u, so mark u as visited

Adjacency List

  • $O(|E| + |V|)$ since you’re visiting all edges and vertices.
  • The loop at 2. takes $O(|V|)$
  • The loop at 4. takes $O(|E_u|)$
    $$\sum\limits_{u=1}^{|V|} \left(\left(\sum\limits_{v=1}^{|E_u|} O(1)\right) + O(1)\right) = |V| \cdot( |E_u| \cdot O(1) + O(1)) = |V| \cdot |E_u| + |V| = |E| + |V|\\$$

Adjacency Matrix

  • $O(|V|^2)$ since you have to check every vertex to find an edge.
  • For adjacency matrices, step 4. takes $O(|V|)$ since looking for all adjacent vertices takes $O(|V|)$ $$\sum\limits_{u=1}^{|V|} \left(\left(\sum\limits_{v=1}^{|V|} O(1)\right) + O(1)\right) = |V| \cdot( |V| \cdot O(1) + O(1)) = |V| \cdot |V| + |V| = |V|^2 + |V|$$
    $$ = O(|V|^2)$$
  • Discover order list: Order of visited nodes
  • Finish order list: Order that vertices are finished (backtrack/return)
Mark the current vertex, u, visited, and put it in discover order list
for each vertex, v, adjacent to the current vertex, u
  if v has not been visited
    set parent of v to u
    recursively apply this algorithm starting at v
mark u finished and enter u into the finish order list
  • Also $O(|E| + |V|)$ for adjacency lists.

  • Also $O(|V|^2)$ for adjacency matrices.

  • Probably better than bfs if you know what you’re searching for is deeper in the graph from the starting point

Dijkstra’s Algorithm

  • S: List with all the vertices that have been processed
  • V-S: List with all the vertices that haven’t been processed completely yet
  • d: Set with the currently known minimum distance for a vertex from the start
  • p: Set with the predecessor of the vertex to get to that vertex
    // Set up the V-S and S lists as well as the p[v] and d[v] sets
 1. Initialize S with the start vertex, s, with V-S with the remaining vertices
 2. for all v in V-S
 3.   Set p[v] to s
 4.   if there is an edge (s, v)
 5.     Set d[v] to w(s, v)
 6.   else
 7.     set d[v] to infinity
 8.   
 9. while V-S is not empty
10.   for all u in V-S, find the smallest d[u]
11.   Remove u from V-S and add u to S
12.   for all v adjacent to u
13.     if d[u] + w(u, v) is less than d[v]
14.       Set d[v] to d[u] + w(u, v)
15.       Set p[v] to u
  • The line at 1. takes $|V|$ steps to go through each vertex and put them in either V-S or S

  • The loop at 2. is executed $|V-1|$ times since all the vertexes except the start are in V-S

  • The loop at 9. also takes $|V-1|$ for similar reasons as 2.

  • Line 10. takes $O(|V|)$ to look through V-S

  • Line 12. takes $O(|E_u)$ to look through the adjacent vertices

  • Total time complexity of dijkstra’s is $O(|V|^2)$ $$O(|V|) + O(|V|) + \left(\sum\limits_{u=1}^{|V|-1} \left(|V| + \sum\limits_{v=1}^{|E_u|} O(1) \right)\right) = 2\cdot O(|V|) + O(|V|^2) + O(|E|) $$ $$ = O(|V|^2)$$ Note that $O(|E|)$ is between $O(1)$ and $O(|V|^2)$, which is why $O(|V|^2) + O(|E|) = O(|V|^2)$.

Topological Sort

If you have a DAG (Directed Acyclic Graph), you can find an ordering so that for every directed edge uv from vertex u to vertex v, u comes before v.
Used for scheduling (e.g. class planning). Note that more than one valid topological sort can exist for a DAG.

Kahn’s Algorithm

First, find a list of “start nodes” which have no incoming edges and insert them into a set S; at least one such node must exist in a non-empty acyclic graph. Then:

L ← Empty list that will contain the sorted elements
S ← Set of all nodes with no incoming edge

while S is not empty do
    remove a node n from S
    add n to L
    for each node m with an edge e from n to m do
        remove edge e from the graph
        if m has no other incoming edges then
            insert m into S

if graph has edges then
    return error   (graph has at least one cycle)
else 
    return L   (a topologically sorted order)

If the graph is a DAG, a solution will be contained in the list L (the solution is not necessarily unique). Otherwise, the graph must have at least one cycle and therefore a topological sort is impossible.

  • Time complexity: $O(|V| + |E|)$
  • Analysis is similar to breadth first search