Editing Segment tree
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− | The '''segment tree''' is a highly versatile | + | The '''segment tree''' is a highly versatile data structure, based upon the [[Divide and conquer|divide-and-conquer]] paradigm, which can be thought of as a tree of intervals of an underlying array, constructed so that queries on ranges of the array as well as modifications to the array's elements may be efficiently performed. |
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==Structure== | ==Structure== | ||
[[File:Segtree_92631507.png|200px|thumb|right|This segment tree.]] | [[File:Segtree_92631507.png|200px|thumb|right|This segment tree.]] | ||
− | Suppose that we use the function defined above to evaluate <math>f(1,N)</math>, where <math>N</math> is the number of elements in the array. When <math>N</math> is large, this recursive call has two "children", one of which is the recursive call <math>f | + | Suppose that we use the function defined above to evaluate <math>f(1,N)</math>, where <math>N</math> is the number of elements in the array. When <math>N</math> is large, this recursive call has two "children", one of which is the recursive call <math>f(1,\lfloor\frac{N+1}{2}\rfloor)</math>, and the other one of which is <math>f(\lfloor\frac{N+1}{2}\rfloor+1,N)</math>. Each of these children will then have two children of its own, and so on, down until the base case is reached. If we represent these recursive calls with a tree structure, the call <math>f(1,N)</math> would be the root, it would have two children, each child would have two more children, and so on; the base cases would be the leaves of the tree. We are now ready to specify the structure of the segment tree: |
* it is a binary tree which represents some underlying array; | * it is a binary tree which represents some underlying array; | ||
* each node is associated with some interval of the array and contains the value(s) of one or more functions of the elements in that interval; | * each node is associated with some interval of the array and contains the value(s) of one or more functions of the elements in that interval; | ||
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===Query=== | ===Query=== | ||
+ | Hai!!! Mesa happie to se yousa! | ||
[[File:Segtree_query_92631587.png|200px|thumb|right|Only the two nodes marked in yellow must be accessed to find the minimum of the elements corresponding to the leaves marked in grey.]] | [[File:Segtree_query_92631587.png|200px|thumb|right|Only the two nodes marked in yellow must be accessed to find the minimum of the elements corresponding to the leaves marked in grey.]] | ||
To ''query'' a segment tree is to use it to determine a function of a range in the underlying array (in this case, the minimum element of that range). The execution of a query is more complex than the execution of an update and will be illustrated by example. Suppose we wish to know the minimum element between the first and sixth, inclusive. We shall represent this query as <math>f(1,6)</math>. Each node in the segment tree contains the minimum in some interval: for example, the root node contains <math>f(1,8)</math>, its left child <math>f(1,4)</math>, its right <math>f(5,8)</math>, and so on, with each leaf containing <math>f(x,x)</math> for some <math>x</math>. There is no node that contains <math>f(1,6)</math>, but we notice that <math>f(1,6) = \min(f(1,4),f(5,6))</math>, and that there ''are'' nodes in the segment tree containing those two values (shown in yellow). (This expression for <math>f(1,6)</math> is not the one given by the definition of <math>f</math>, but it is fairly clear that <math>f(x,y) = \min(f(x,z),f(z+1,y))</math> where <math>x \le z < y</math>, regardless of the actual value of <math>z</math>.)<br/> | To ''query'' a segment tree is to use it to determine a function of a range in the underlying array (in this case, the minimum element of that range). The execution of a query is more complex than the execution of an update and will be illustrated by example. Suppose we wish to know the minimum element between the first and sixth, inclusive. We shall represent this query as <math>f(1,6)</math>. Each node in the segment tree contains the minimum in some interval: for example, the root node contains <math>f(1,8)</math>, its left child <math>f(1,4)</math>, its right <math>f(5,8)</math>, and so on, with each leaf containing <math>f(x,x)</math> for some <math>x</math>. There is no node that contains <math>f(1,6)</math>, but we notice that <math>f(1,6) = \min(f(1,4),f(5,6))</math>, and that there ''are'' nodes in the segment tree containing those two values (shown in yellow). (This expression for <math>f(1,6)</math> is not the one given by the definition of <math>f</math>, but it is fairly clear that <math>f(x,y) = \min(f(x,z),f(z+1,y))</math> where <math>x \le z < y</math>, regardless of the actual value of <math>z</math>.)<br/> |