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<table class="table table-bordered table-striped" id="legend">
<tbody>
<tr>
<td><code class="green">Good</code></td>
<td><code class="yellow">Fair</code></td>
<td><code class="red">Poor</code></td>
</tr>
</tbody>
</table>
<h2 id="searching">Searching</h2>
<table class="table table-bordered table-striped">
<thead>
<tr>
<th>Algorithm</th>
<th>Data Structure</th>
<th colspan="2">Time Complexity</th>
<th colspan="3">Space Complexity</th>
</tr>
<tr>
<th></th>
<th></th>
<th>Average</th>
<th>Worst</th>
<th>Worst</th>
</tr>
</thead>
<tbody>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Depth-first_search">Depth First Search (DFS)</a></td>
<td>Graph of |V| vertices and |E| edges</td>
<td><code>-</code></td>
<td><code class="green">O(|E| + |V|)</code></td>
<td><code class="green">O(|V|)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Breadth-first_search">Breadth First Search (BFS)</a></td>
<td>Graph of |V| vertices and |E| edges</td>
<td><code>-</code></td>
<td><code class="green">O(|E| + |V|)</code></td>
<td><code class="green">O(|V|)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Binary_search_algorithm">Binary search</a></td>
<td>Sorted array of n elements</td>
<td>
<code class="green">O(log(n))</code>
</td>
<td>
<code class="green">O(log(n))</code>
</td>
<td>
<code class="green">O(1)</code>
</td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Brute-force_search">Linear (Brute Force)</a></td>
<td>Array</td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="green">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Dijkstra's_algorithm">Shortest path by Dijkstra,<br>using a Min-heap as priority queue</a></td>
<td>Graph with |V| vertices and |E| edges</td>
<td><code class="yellow">O((|V| + |E|) log |V|)</code></td>
<td><code class="yellow">O((|V| + |E|) log |V|)</code></td>
<td><code class="yellow">O(|V|)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Dijkstra's_algorithm">Shortest path by Dijkstra,<br>using an unsorted array as priority queue</a></td>
<td>Graph with |V| vertices and |E| edges</td>
<td><code class="yellow">O(|V|<sup>2</sup>)</code></td>
<td><code class="yellow">O(|V|<sup>2</sup>)</code></td>
<td><code class="yellow">O(|V|)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm">Shortest path by Bellman-Ford</a></td>
<td>Graph with |V| vertices and |E| edges</td>
<td><code class="yellow">O(|V||E|)</code></td>
<td><code class="yellow">O(|V||E|)</code></td>
<td><code class="yellow">O(|V|)</code></td>
</tr>
</tbody>
</table>
<h2 id="sorting">Sorting</h2>
<table class="table table-bordered table-striped">
<thead>
<tr>
<th>Algorithm</th>
<th>Data Structure</th>
<th colspan="3">Time Complexity</th>
<th colspan="3">Worst Case Auxiliary Space Complexity</th>
</tr>
<tr>
<th></th>
<th></th>
<th>Best</th>
<th>Average</th>
<th>Worst</th>
<th>Worst</th>
</tr>
</thead>
<tbody>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Quicksort">Quicksort</a></td>
<td>Array</td>
<td><code class="yellow">O(n log(n))</code></td>
<td><code class="green">O(n log(n))</code></td>
<td><code class="red">O(n<sup>2</sup>)</code></td>
<td><code class="yellow">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Merge_sort">Mergesort</a></td>
<td>Array</td>
<td><code class="yellow">O(n log(n))</code></td>
<td><code class="green">O(n log(n))</code></td>
<td><code class="green">O(n log(n))</code></td>
<td><code class="red">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Heapsort">Heapsort</a></td>
<td>Array</td>
<td><code class="yellow">O(n log(n))</code></td>
<td><code class="green">O(n log(n))</code></td>
<td><code class="green">O(n log(n))</code></td>
<td><code class="green">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Bubble_sort">Bubble Sort</a></td>
<td>Array</td>
<td><code class="green">O(n)</code></td>
<td><code class="red">O(n<sup>2</sup>)</code></td>
<td><code class="red">O(n<sup>2</sup>)</code></td>
<td><code class="green">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Insertion_sort">Insertion Sort</a></td>
<td>Array</td>
<td><code class="green">O(n)</code></td>
<td><code class="red">O(n<sup>2</sup>)</code></td>
<td><code class="red">O(n<sup>2</sup>)</code></td>
<td><code class="green">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Selection_sort">Select Sort</a></td>
<td>Array</td>
<td><code class="red">O(n<sup>2</sup>)</code></td>
<td><code class="red">O(n<sup>2</sup>)</code></td>
<td><code class="red">O(n<sup>2</sup>)</code></td>
<td><code class="green">O(1)</code></td>
</tr>
<tr>
<td><a rel="tooltip" title="Only for integers with range k" href="http://en.wikipedia.org/wiki/Bucket_sort">Bucket Sort</a></td>
<td>Array</td>
<td><code class="green">O(n+k)</code></td>
<td><code class="green">O(n+k)</code></td>
<td><code class="red">O(n<sup>2</sup>)</code></td>
<td><code class="yellow">O(nk)</code></td>
</tr>
<tr>
<td><a rel="tooltip" title="Constant number of digits 'k'" href="http://en.wikipedia.org/wiki/Radix_sort">Radix Sort</a></td>
<td>Array</td>
<td><code class="green">O(nk)</code></td>
<td><code class="green">O(nk)</code></td>
<td><code class="green">O(nk)</code></td>
<td><code class="yellow">O(n+k)</code></td>
</tr>
</tbody>
</table>
<h2 id="data-structures">Data Structures</h2>
<table class="table table-bordered table-striped">
<thead>
<tr>
<th>Data Structure</th>
<th colspan="8">Time Complexity</th>
<th>Space Complexity</th>
</tr>
<tr>
<th></th>
<th colspan="4">Average</th>
<th colspan="4">Worst</th>
<th>Worst</th>
</tr>
<tr>
<th></th>
<th>Indexing</th>
<th>Search</th>
<th>Insertion</th>
<th>Deletion</th>
<th>Indexing</th>
<th>Search</th>
<th>Insertion</th>
<th>Deletion</th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Array_data_structure">Basic Array</a></td>
<td><code class="green">O(1)</code></td>
<td><code class="red">O(n)</code></td>
<td><code>-</code></td>
<td><code>-</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="red">O(n)</code></td>
<td><code>-</code></td>
<td><code>-</code></td>
<td><code class="yellow">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Dynamic_array">Dynamic Array</a></td>
<td><code class="green">O(1)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="yellow">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Singly_linked_list#Singly_linked_lists">Singly-Linked List</a></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="yellow">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Doubly_linked_list">Doubly-Linked List</a></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="yellow">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Skip_list">Skip List</a></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n log(n))</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Hash_table">Hash Table</a></td>
<td><code>-</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="green">O(1)</code></td>
<td><code>-</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="yellow">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Binary_search_tree">Binary Search Tree</a></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="yellow">O(n)</code></td>
</tr>
<tr>
<td><a href="https://en.wikipedia.org/wiki/Cartesian_tree">Cartresian Tree</a></td>
<td><code>-</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code>-</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="red">O(n)</code></td>
<td><code class="yellow">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/B_tree">B-Tree</a></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="yellow">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Red-black_tree">Red-Black Tree</a></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="yellow">O(n)</code></td>
</tr>
<tr>
<td><a href="https://en.wikipedia.org/wiki/Splay_tree">Splay Tree</a></td>
<td><code>-</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code>-</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="yellow">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/AVL_tree">AVL Tree</a></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="green">O(log(n))</code></td>
<td><code class="yellow">O(n)</code></td>
</tr>
</tbody>
</table>
<h2 id="heaps">Heaps</h2>
<table class="table table-bordered table-striped">
<thead>
<tr>
<th>Heaps</th>
<th colspan="7">Time Complexity</th>
</tr>
<tr>
<th></th>
<th>Heapify</th>
<th>Find Max</th>
<th>Extract Max</th>
<th>Increase Key</th>
<th>Insert</th>
<th>Delete</th>
<th>Merge</th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Linked_list">Linked List (sorted)</a></td>
<td><code>-</code></td> <!-- heapify -->
<td><code class="green">O(1)</code></td> <!-- Find max -->
<td><code class="green">O(1)</code></td> <!-- Extract max -->
<td><code class="red">O(n)</code></td> <!-- Increase -->
<td><code class="red">O(n)</code></td> <!-- insert -->
<td><code class="green">O(1)</code></td> <!-- delete -->
<td><code class="red">O(m+n)</code></td> <!-- merge -->
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Linked_list">Linked List (unsorted)</a></td>
<td><code>-</code></td> <!-- heapify -->
<td><code class="red">O(n)</code></td> <!-- Find max -->
<td><code class="red">O(n)</code></td> <!-- Extract max -->
<td><code class="green">O(1)</code></td> <!-- Increase -->
<td><code class="green">O(1)</code></td> <!-- insert -->
<td><code class="green">O(1)</code></td> <!-- delete -->
<td><code class="green">O(1)</code></td> <!-- merge -->
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Binary_heap">Binary Heap</a></td>
<td><code class="yellow">O(n)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="red">O(m+n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Binomial_heap">Binomial Heap</a></td>
<td><code>-</code></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="yellow">O(log(n))</code></td>
<td><code class="yellow">O(log(n))</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Fibonacci_heap">Fibonacci Heap</a></td>
<td><code>-</code></td>
<td><code class="green">O(1)</code></td>
<td><code rel="tooltip" title="Amortized" class="yellow">O(log(n))*</code></td>
<td><code rel="tooltip" title="Amortized" class="green">O(1)*</code></td>
<td><code class="green">O(1)</code></td>
<td><code rel="tooltip" title="Amortized" class="yellow">O(log(n))*</code></td>
<td><code class="green">O(1)</code></td>
</tr>
</tbody>
</table>
<h2 id="graphs">Graphs</h2>
<table class="table table-bordered table-striped">
<tbody>
<tr>
<th>Node / Edge Management</th>
<th>Storage</th>
<th>Add Vertex</th>
<th>Add Edge</th>
<th>Remove Vertex</th>
<th>Remove Edge</th>
<th>Query</th>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Adjacency_list">Adjacency list</a></td>
<td><code class="yellow">O(|V|+|E|)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="yellow">O(|V| + |E|)</code></td>
<td><code class="yellow">O(|E|)</code></td>
<td><code class="yellow">O(|V|)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Incidence_list">Incidence list</a></td>
<td><code class="yellow">O(|V|+|E|)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="yellow">O(|E|)</code></td>
<td><code class="yellow">O(|E|)</code></td>
<td><code class="yellow">O(|E|)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Adjacency_matrix">Adjacency matrix</a></td>
<td><code class="red">O(|V|<sup>2</sup>)</code></td>
<td><code class="red">O(|V|<sup>2</sup>)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="red">O(|V|<sup>2</sup>)</code></td>
<td><code class="green">O(1)</code></td>
<td><code class="green">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Incidence_matrix">Incidence matrix</a></td>
<td><code class="red">O(|V| ⋅ |E|)</code></td>
<td><code class="red">O(|V| ⋅ |E|)</code></td>
<td><code class="red">O(|V| ⋅ |E|)</code></td>
<td><code class="red">O(|V| ⋅ |E|)</code></td>
<td><code class="red">O(|V| ⋅ |E|)</code></td>
<td><code class="yellow">O(|E|)</code></td>
</tr>
</tbody>
</table>
<h2 id="growth">Notation for asymptotic growth</h2>
<table class="table table-bordered table-striped">
<tr>
<th>letter</th>
<th>bound</th>
<th>growth</th>
</tr>
<tr>
<td>(theta) Θ</td>
<td>upper and lower, tight<a href="#footnote-1"><sup>[1]</sup></a></td>
<td>equal<a href="#footnote-1"><sup>[2]</sup></a></td>
</tr>
<tr>
<td>(big-oh) O</td>
<td>upper, tightness unknown</td>
<td>less than or equal<a href="#footnote-1"><sup>[3]</sup></a></td>
</tr>
<tr>
<td>(small-oh) o</td>
<td>upper, not tight</td>
<td>less than</td>
</tr>
<tr>
<td>(big omega) Ω</td>
<td>lower, tightness unknown</td>
<td>greater than or equal</td>
</tr>
<tr>
<td>(small omega) ω</td>
<td>lower, not tight</td>
<td>greater than</td>
</tr>
</table>
<p id="footnote-1">[1] Big O is the upper bound, while Omega is the lower bound. Theta requires both Big O and Omega, so that's why it's referred to as a tight bound (it must be both the upper and lower bound). For example, an algorithm taking Omega(n log n) takes at least n log n time but has no upper limit. An algorithm taking Theta(n log n) is far preferential since it takes AT LEAST n log n (Omega n log n) and NO MORE THAN n log n (Big O n log n).<sup><a href="http://stackoverflow.com/a/464081/412916">SO</a></sup></p>
<p id="footnote-2">[2] f(x)=Θ(g(n)) means f (the running time of the algorithm) grows exactly like g when n (input size) gets larger. In other words, the growth rate of f(x) is asymptotically proportional to g(n).</p>
<p id="footnote-1">[3] Same thing. Here the growth rate is no faster than g(n). big-oh is the most useful because represents the worst-case behavior.</p>
In short, if algorithm is __ then its performance is __
<table class="table table-bordered table-striped">
<tr>
<th>algorithm</th>
<th>performance</th>
</tr>
<tr>
<td>o(n)</td>
<td>< n</td>
</tr>
<tr>
<td>O(n)</td>
<td>≤ n</td>
</tr>
<tr>
<td>Θ(n)</td>
<td>= n</td>
</tr>
<tr>
<td>Ω(n)</td>
<td>≥ n</td>
</tr>
<tr>
<td>ω(n)</td>
<td>> n</td>
</tr>
</table>