98. Data Structures – Heaps and Tries

Released Monday, 21st January 2019
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We dig into heaps and tries as Allen gives us an up to date movie review while Joe and Michael compare how the bands measure up.

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Trie as you may …



Heaps



What is it?


  • Heaps are specialized binary trees that keep data sorted-ish.
  • This makes them slower for lookups, but faster for insertions.
  • They are also really good at getting their roots plucked.
  • Wikipedia list 18 different types of heaps, but we focus on Min/Max heaps (particularly max).
  • Sorted-ish?
    • Heaps are very similar to binary search tree (BST), they have two values – but unlike a BST, the children are always less than the parent (or more in a min tree).
    • This means that your root node is always the biggest value in the whole tree.
    • Because heaps aren’t so strict on the ordering, it’s quicker, O(log n), to insert data than it is into a sorted array.
    • It’s also quicker on average to insert into than a BST, in both cases the insertion time comes down to the height of the tree but we can cheaply keep the heap balanced – no straight lines here!
    • The downside is that heaps are slower for lookups, because you potentially have to look at every node in the tree.
    • Finally, the killer feature – it’s really efficient to remove the root node.


How does it work?


  • Heaps are typically stored in an array. Because it’s a binary tree you can quickly calculate a node’s children based on its index.
  • Downside is you need to pre-allocate memory, and either have to set a max size or use a dynamic array data structure that will grow as needed.
  • Adding to the heap has cool names like “up-heap”, “bubble up”, “trickle up”.
    • Add your node to the bottom level (easy in an array, just shove it in the first open value).
    • If the new node is less than its parent, swap and repeat.
  • Extraction, downheap, trickle-down, bubble-down, etc.
    • Move the last filled value in the array to 0.
    • If this node is greater than one of its children, swap and repeat.


Pros


  • Fast insert O(1) on average, O(log n) for worst case.
  • Fast removal of root node.


Cons


  • Slow search O(n).


When to use?


  • When you need to quickly insert data, and you only care about the min or max value.
  • Don’t use a heap when you need to search for arbitrary values.
  • Priority Queues (BFS).


Tries



What is it?


  • Tries, (technically) pronounced “tree” as in retrieval, aka digital tree, (compressed) radix tree, prefix tree.
  • They are specialized tree where the nodes don’t matter.
  • Instead, the “key” or “value” or “payload” is associated with the edges, and the nodes just exist for convenience.
  • The true values in the tree only exist on the leaves, and siblings share what is called a common prefix.


How does it work?


  • English Spellchecker example:
    • English is a mess, every rule has an exception – q is not always followed by u, i is not always before e, y isn’t always a vowel.
    • You need to store every english word in order to implement a spell checker (~1M depending on how you count).
    • You could keep a hash table, or a big sorted array then scan the document word by word to look to see if it’s valid.
      • This should be fast enough too, worst case for hash or sorted array is O(log n) (worst case 20 operations to find a word in the WORST case!) but how much RAM would you need to keep 1M words in memory…
    • However, if you look at the data you’ll see that a lot of words are really similar – code, codes, coder, coders, coding, …
    • We can do better with tries!!
    • Let’s start with an empty trie, and every time we add a word, lets have an edge represent one letter of the word.
    • When we add the word “code” we get an edge for c, o, d, and e – when we add “coder” …we already have edges for 4 of the letters, so we just add an edge for the r.
    • When we stored each of those coding words in an array, we ended up storing 26 characters, using a trie: 10.
    • Sure, we saved some space – but think about what it takes to see if a word exists? Checking for “code” just means looking for a branch for each letter – the number of comparisons is equal to the number of letters.
    • To make things even better, it’s actually really easy to offer auto-complete now – as the user types a letter, we can take another step down the trie to see if it exists … and even follow the tree down to show what words we think you might be typing.
  • While the example used natural language, tries are also useful for permutations, digits, or even binary – basically anything that has a lot of identical prefix data.
  • Note there is actually a better data structure for an autocomplete/spell checker: Deterministic Acyclic Finite State Automaton (DAFSA for short)!
  • Tries also compress really well, like combining any nodes that only have one child – in the spellchecker example, imagine that “q” was always followed by a “u” – then you could combine to a “qu” node.


Pros


  • Often used in place of a hash.
    • Faster worse case lookup than an imperfect hash table.
    • No messy/bad hash functions to deal with.
    • No messy/bad collision strategies to deal with.
    • Easy to provide an ordered listing of keys with no additional data structures.


Cons


  • Trie lookups can be slower than hash lookups, particularly if the random-access time of the medium is bad.
  • Not good when there would be long meaningless chains (like 2.3333333333334).
  • Can potentially take more space than a hash table (imagine having the pointer over head for each character where there isn’t a lot of overhead, as opposed to a single pointer to a string).


When to use?


  • Fast insert, fast and efficient sorting, lots of duplicated data…like in the bowels of search engines, or spell checkers/autocomplete.


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