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10.1: Hashing

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  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    To improve the performance of MyLinearMap, we’ll write a new class, called MyBetterMap, that contains a collection of MyLinearMap objects. It divides the keys among the embedded maps, so the number of entries in each map is smaller, which speeds up findEntry and the methods that depend on it.

    Here’s the beginning of the class definition:

    public class MyBetterMap<K, V> implements Map<K, V> {
        protected List<MyLinearMap<K, V>> maps;
        public MyBetterMap(int k) {
        protected void makeMaps(int k) {
            maps = new ArrayList<MyLinearMap<K, V>>(k);
            for (int i=0; i<k; i++) {
                maps.add(new MyLinearMap<K, V>());

    The instance variable, maps, is a collection of MyLinearMap objects. The constructor takes a parameter, k, that determines how many maps to use, at least initially. Then makeMaps creates the embedded maps and stores them in an ArrayList.

    Now, the key to making this work is that we need some way to look at a key and decide which of the embedded maps it should go into. When we put a new key, we choose one of the maps; when we get the same key, we have to remember where we put it.

    One possibility is to choose one of the sub-maps at random and keep track of where we put each key. But how should we keep track? It might seem like we could use a Map to look up the key and find the right sub-map, but the whole point of the exercise is to write an efficient implementation of a Map. We can’t assume we already have one.

    A better approach is to use a hash function, which takes an Object, any Object, and returns an integer called a hash code. Importantly, if it sees the same Object more than once, it always returns the same hash code. That way, if we use the hash code to store a key, we’ll get the same hash code when we look it up.

    In Java, every Object provides a method called hashCode that computes a hash function. The implementation of this method is different for different objects; we’ll see an example soon.

    Here’s a helper method that chooses the right sub-map for a given key:

    protected MyLinearMap<K, V> chooseMap(Object key) {
        int index = 0;
        if (key != null) {
            index = Math.abs(key.hashCode()) % maps.size();
        return maps.get(index);

    If key is null, we choose the sub-map with index 0, arbitrarily. Otherwise we use hashCode to get an integer, apply Math.abs to make sure it is non-negative, then use the remainder operator, %, which guarantees that the result is between 0 and maps.size()-1. So index is always a valid index into maps. Then chooseMap returns a reference to the map it chose.

    We use chooseMap in both put and get, so when we look up a key, we get the same map we chose when we added the key. At least, we should — I’ll explain a little later why this might not work.

    Here’s my implementation of put and get:

    public V put(K key, V value) {
        MyLinearMap<K, V> map = chooseMap(key);
        return map.put(key, value);
    public V get(Object key) {
        MyLinearMap<K, V> map = chooseMap(key);
        return map.get(key);

    Pretty simple, right? In both methods, we use chooseMap to find the right sub-map and then invoke a method on the sub-map. That’s how it works; now let’s think about performance.

    If there are n entries split up among k sub-maps, there will be \( n \div k \) entries per map, on average. When we look up a key, we have to compute its hash code, which takes some time, then we search the corresponding sub-map.

    Because the entry lists in MyBetterMap are k times shorter than the entry list in MyLinearMap, we expect the search to be k times faster. But the run time is still proportional to n, so MyBetterMap is still linear. In the next exercise, you’ll see how we can fix that.

    This page titled 10.1: Hashing is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Allen B. Downey (Green Tea Press) .

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