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    <p>If your i,k,...,z are arbitrary, then no algorithm with a time constraint of O(1) and a reasonable space constraint of &lt; O(N) comes to mind. If there is some regularity in your bounds, then of course you can put that into use: e.g. if each range contains an even number of elements, then a simple solution be to have a lookup table that is acccessed using [i_N/2]. In the general case, you would need to have a matching folding-function for your number 0...N - one that satisfies the constraint k-mapping(i) != k-mapping(i+1) => folding-function(i) != folding-function(i+1).</p> <p>If only some elements in your map are accessed frequently, you could also build a cache for the mapping, leading to O(1) in the cache-hit case and O(lookup-function) for the cache-miss. Of course, the cache lookup would impact the time spent for each element, even in case of a cache-miss, but it would not affect the O-complexity of your algorithm.</p> <h3>Edit</h3> <p>Basically what you want to implement can be expressed as a Pascal case statement with sets as labels:</p> <pre><code>case n of 0..i-1: value:=0; i..i+k-1: value:=1; // ... i+k+...+z+2..N-1: value:=K; end </code></pre> <p>For this problem, a Google search indeed turns out an interesting paper:</p> <blockquote> <p>This paper descripts a new scheme for building static search trees, using multiway radix search trees. [...] We show that, for sparse case sets, the method produces faster code on average than existing methods, requiring <em>O</em>(1) time with a small constant for the average search.</p> <p><em>Erlingsson, Krishnamoorthy, Raman</em>: <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.48.2872&amp;rep=rep1&amp;type=pdf" rel="nofollow">Lucid and Efficient Case Analysis</a></p> </blockquote> <p>Alas, you don't have a sparse case set, but as far as I can see, the algorithm should require only minimal adaption for dense case sets that have lots of consecutive results.</p> <p>For the general problem of creating efficient code for case statements, Zhao and Amarla have an <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.69.7532&amp;rep=rep1&amp;type=pdf" rel="nofollow">interesting paper that uses profiling</a> to achieve something similar to using a cache. Their paper is also interesting for the related work section, which refers to a paper from <em>Kannan and Proebsting</em> (<em>Correction to producing good code for the case statement.</em>) which requires an O(n^2) setup time for clustering. However, I have not access to this paper, and it sounds like that this would only produce an extremely optimized search tree, therefore leading to a runtime of O(log(n)).</p>
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    1. COBy now, I'm seriously tempted to believe the same (i.e: there's no O(1) algorithm with a "small" space constraint). If no one comes with a good answer, I'll take yours as a right answer.
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    2. COGreat Answer! Very useful paper too! Thanks!
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