Pf congres20110917 data-structures

SPL Data Structures and their Complexity Jurri¨n Stutterheim e September 17, 20111

1. Introduction2

This presentation §1 Understand what data structures are How they are represented internally How “fast” each one is and why that is3

Data structures §1 Classes that oﬀer the means to store and retrieve data, possibly in a particular order Implementation is (often) optimised for certain use cases array is PHP’s oldest and most frequently used data structure PHP 5.3 adds support for several others4

Current SPL data structures §1 SplDoublyLinkedList SplStack SplQueue SplHeap SplMaxHeap SplMinHeap SplPriorityQueue SplFixedArray SplObjectStorage5

Why care? §1 Using the right data structure in the right place could improve performance Already implemented and tested: saves work Can add a type hint in a function deﬁnition Adds semantics to your code6

Algorithmic complexity §1 We want to be able to talk about the performance of the data structure implementation Running speed (time complexity) Space consumption (space complexity) We describe complexity in terms of input size, which is machine and programming language independent7

Example §1 for ($i = 0; $i < $n; $i++) { for ($j = 0; $j < $n; $j++) { echo ’tick’; } } For some n, how many times is “tick” printed? I.e. what is the time complexity of this algorithm?8

Example §1 for ($i = 0; $i < $n; $i++) { for ($j = 0; $j < $n; $j++) { echo ’tick’; } } For some n, how many times is “tick” printed? I.e. what is the time complexity of this algorithm? n2 times8

Talking about complexity §1 Pick a function to act as boundary for the algorithm’s complexity Worst-case Denoted O (big-Oh) “My algorithm will not be slower than this function” Best-case Denoted Ω (big-Omega) “My algorithm will at least be as slow as this function” If they are the same, we write Θ (big-Theta) In example: both cases are n2 , so the algorithm is in Θ(n2 )9

Visualized §110

Example 2 §1 for ($i = 0; $i < $n; $i++) { if ($myBool) { for ($j = 0; $j < $n; $j++) { echo ’tick’; } } } What is the time complexity of this algorithm?11

Example 2 §1 for ($i = 0; $i < $n; $i++) { if ($myBool) { for ($j = 0; $j < $n; $j++) { echo ’tick’; } } } What is the time complexity of this algorithm? O(n2 ) Ω(n) (if $myBool is false) No Θ!11

We can be a bit sloppy §1 for ($i = 0; $i < $n; $i++) { if ($myBool) { for ($j = 0; $j < $n; $j++) { echo ’tick’; } } } We describe algorithmic behaviour as input size grows to inﬁnity constant factors and smaller terms don’t matter too much E.g. 3n2 + 4n + 1 is in O(n2 )12

Other functions §1 for ($i = 0; $i < $n; $i++) { for ($j = 0; $j < $n; $j++) { echo ’tick’; } } for ($i = 0; $i < $n; $i++) { echo ’tock’; } This algorithm is still in Θ(n2 ).13

Bounds §1 Figure: Order relations1 114 Taken from Cormen et al. 2009

Complexity Comparison §1 3 10 Superexponential Factorial Exponential Quadratic 102 Linear 1 10 Logarithmic 100 101 Constant: 1, logarithmic: lg n, linear: n, quadratic: n2 , exponential: 2n , factorial: n!, super-exponential: nn15

In numbers §1 Approximate growth for n = 50: 1 1 lg n 5.64 n 50 n2 2500 n3 12500 2n 1125899906842620 n! 3.04 ∗ 1064 nn 8.88 ∗ 108416

Some more notes on complexity §1 Constant time is written 1, but goes for any constant c Polynomial time contains all functions in nc for some constant c Everything in this presentation will be in polynomial time17

2. SPL Data Structures18

Credit where credit is due §2 The ﬁrst three pictures in this section are from Wikipedia19

SplDoublyLinkedList §2 12 99 37 Superclass of SplStack and SplQueue SplDoublyLinkedList is not truly a doubly linked list; it behaves like a hashtable20

SplDoublyLinkedList §2 12 99 37 Superclass of SplStack and SplQueue SplDoublyLinkedList is not truly a doubly linked list; it behaves like a hashtable Usual doubly linked list time complexity Append/prepend to available node in Θ(1) Lookup by scanning in O(n) Access to beginning/end in Θ(1)20

SplDoublyLinkedList §2 12 99 37 Superclass of SplStack and SplQueue SplDoublyLinkedList is not truly a doubly linked list; it behaves like a hashtable Usual doubly linked list time complexity Append/prepend to available node in Θ(1) Lookup by scanning in O(n) Access to beginning/end in Θ(1) SplDoublyLinkedList time complexity Insert/delete by index in Θ(1) Lookup by index in Θ(1) Access to beginning/end in Θ(1)20

SplStack §2 Subclass of SplDoublyLinkedList; adds no new operations Last-in, ﬁrst-out (LIFO) Pop/push value from/on the top of the stack in Θ(1) Push Pop21

SplQueue §2 Subclass of SplDoublyLinkedList; adds enqueue/dequeue operations First-in, ﬁrst-out (FIFO) Read/dequeue element from front in Θ(1) Enqueue element to the end in Θ(1) Enqueue Dequeue22

Short excursion: trees §2 100 19 36 17 3 25 1 2 7 Consists of nodes (vertices) and directed edges Each node always has in-degree 1 Except the root: always in-degree 0 Previous property implies there are no cycles Binary tree: each node has at most two child-nodes23

SplHeap, SplMaxHeap and SplMinHeap §2 100 19 36 17 3 25 1 2 7 A heap is a tree with the heap property : for all A and B, if B is a child node of A, then val(A) val(B) for a max-heap: SplMaxHeap val(A) val(B) for a min-heap: SplMinHeap Where val(A) denotes the value of node A24

Heaps contd. §2 SplHeap is an abstract superclass Implemented as binary tree Access to root element in Θ(1) Insertion/deletion in O(lg n)25

SplPriorityQueue §2 Variant of SplMaxHeap: for all A and B, if B is a child node of A, then prio(A) prio(B) Where prio(A) denotes the priority of node A26

SplFixedArray §2 Fixed-size array with numerical indices only Eﬃcient OO array implementation No hashing required for keys Can make assumptions about array size Lookup, insertion, deletion in Θ(1) time Resize in Θ(n)27

SplObjectStorage §2 Storage container for objects Insertion, deletion in Θ(1) Veriﬁcation of presence in Θ(1) Missing: set operations Union, intersection, diﬀerence, etc.28

3. Concluding29

Missing in PHP §3 Set data structure Map/hashtable data structure Does SplDoublyLinkedList satisfy this use case? If yes: split it in two separate structures and make SplDoublyLinkedList a true doubly linked list Immutable data structures Allows us to more easily emulate “pure” functions Less bugs in your code due to lack of mutable state30

Closing remarks §3 Use the SPL data structures! Choose them with care Reason about your code’s complexity31

Questions §3 Questions?32

Pf congres20110917 data-structures

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Pf congres20110917 data-structures — Presentation Transcript