To implement the interval-search-exclusive operation, we can add a helper function to the interval tree to recursively search for nodes that match a given interval. Here's a possible implementation:
type intervaltreenode struct func (t *intervaltreenode) searchexactly(i interval) *intervaltreenode if t.ptr != nil &&t.ptr.interval == i if i.start <= t.interval.start }if i.end >= t.interval.end }return nil }Now we can call the interval-search-exclusive operation using the following:
func interval-search-exactly(t *intervaltreenode, i interval) *intervaltreenode else }
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In go, we can solve this problem using binary search. We first find the root node of the interval tree, and then compare the interval of the root node with the given interval. If the interval of the root node is equal to the given compartment, we return the root node. I'm so sorry, I might need to react for a while, give me some time, thank you.
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Here's an algorithm implemented by Go to interval-search-exactly on the interval tree t and a interval i:
type intervaltreenode struct func intervalsearchexactly(t *intervaltreenode, i int) *intervaltreenode returns nil if tmax != i If the high-end of the current node is not equal to the high-end of the query interval, nil if t. is returnedmax != i If the interval of the current node is exactly equal to the query interval of the current node, return the current node return t }
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In order to support the new operation intervalsearch-exactly(t, i), we need to extend the structure and operations of the interval tree. To do this in Go, we need to follow these steps:
1.Defines the data structure of the interval tree, including the definition of the compartments (low and high) and the tree nodes (which may contain pointers to child nodes and compartment information).
2.Implement insertion, deletion, and query operations for interval trees, ensuring that all operations run at o(lgn).
3.Implement the intervalsearch-exactly(t, i) operation, which will search for intervals with exact boundary matches.
Here's a simplified Go implementation that shows how to define the data structure of an interval tree, and how to implement the intervalsearch-exactly operation:
package mainimport ( "fmt"interval defines an interval type interval struct intervaltreenode is a node of the interval treetype intervaltreenode struct newintervaltreenode create a new interval tree nodefunc newintervaltreenode(interval interval) * IntervalTreeNode } IntervalTree is an encapsulation of the interval treeType IntervalTree struct insert Insert a new interval func (T *intervaltree) insert(interval interval) searchexactly to search for an interval func (T *intervaltree) with exact boundary matching) searchexactly(i interval) *intervaltreenode searchexactly recursively search for an exact match of intervals in the interval treefunc searchexactly(node *intervaltreenode, i interval) *intervaltreenode if ilow == node.interval.low &&i.high == node.interval.high selects either the left subtree or the right subtree based on the boundary value of the interval if ilow < node.interval.low else if i.low > node.interval.low if low is the same, further check if ihigh < node.interval.high else if i.high > node.interval.high If high is also the same, then a matching interval is found return node}func main() inserted into the interval treeinsert(interval) // tree.insert(interval) // tree.insert(interval) searches for an exact match between i := interval node := treesearchexactly(i) if node != nil else }
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In the Go language, in order to support the intervalsearch-exclusive operation, you can make the following modifications on the basis of the existing interval tree: Let's assume that we already have a compartment tree structure that implements the basic interval search function. First of all, we need to expand the interval tree node to add a flag to indicate whether the interval represented by the node accurately matches the target interval. Next, a recursive exactMatchSearch function is provided that finds the node of the interval that is exactly the same as the given interval in o(log n) time.
type interval struct type node struct intervalsearch-exactly functionfunc intervalsearchexactly(root *node, i interval) *node if root.interval.low == i.low &&root.interval.high == i.high leftresult := intervalsearchexactly(root.left, i) if leftresult != nil rightresult := intervalsearchexactly(root.right, i) if rightresult != nil return nil Exact Matching Interval Not Found} Initialization function to zero the isexactmatch flag when creating and inserting new compartmentsfunc insertinterval(root **node, interval interval) (optional) When deleting a node, you also need to reset the isexactmatch flag of its ancestor nodefunc deletenode(root ** node, i interval) resets the isexactmatch flag of the ancestor node, func resetancestorexactmatch(root **node, deletedinterval interval) if deletedintervallow <= current.interval.low else }
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In GO, interval trees are usually represented by arrays, where each element corresponds to a node. There are two types of queries that we can use:
1.Query for nodes that are exactly the same as the given interval i. This can be done by starting from the root and recursively checking that the intervals of the child nodes are exactly the same as i. If it's the same, we recursively check that child node. 2.Query the nodes that contain a given interval. This can be done by starting at the root, recursively checking whether the child node's interval contains a given compartment. If it does, we recursively check for that child node. The time complexity of both queries is o(logn). So, if we do a query, the total time complexity is O(logn).
Here's an example of a Go implementation**:
type intervaltree struct type intervaltreenode struct func (it *intervaltree) insert(interval interval) func (it *intervaltree) exactsearch(low, high int) *intervaltreenode func (it *intervaltree) exactsearch(node *intervaltreenode, low, high, result *int) if low <= node.low if node.high <= high }Insert a description here.
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For a given interval tree t and a compartment i, we can support the new operation interval-search-exactly(t, i) by modifying the query process for the interval tree. Here's an example of an implementation using go:Type interval struct type node struct type intervaltree struct Create a new interval treefunc newintervaltree() intervaltree , max: -1} return &intervaltree} Insert an interval func (tree *intervaltree) into the interval tree insert(interval interval) current := tree.root var prev *node finds the location to insert for current != tree.nil &¤t != nil else } to insert a new node newnodeparent = prev if prev == tree.nil else if interval.low < prev.interval.low else updates the maximum value of the parent node, treeupdatemax(newnode)} searches the interval tree for nodes that exactly match the given interval func (tree *intervaltree) intervalsearchexactly(interval interval) *node recursively searches the interval tree for nodes that exactly match the given interval func (tree *intervaltree) intervalsearchexactlyhelper(node *node, interval interval) *node if node.left != tree.nil &&node.left.max >= interval.low else } the maximum value of the update node and its ancestor node: func (tree *intervaltree) updatemax(node *node) node = nodeparent }}func main() t.insert(interval) t.insert(interval) t.insert(interval) i := interval result := t.intervalsearchexactly(i) if result != t.nil else }Insert a description here.
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In go, you can use a custom data structure to represent interval trees and implement intervalsearch-exact operations. Here's an example:
package mainimport ( "fmt"Type interval struct type node struct type intervaltree struct Create a new nodefunc newnode(interval interval) *node } Insert node into the interval tree (based on the red-black tree)func (it *intervaltree) insert(node *node) else }node.p = y sets the parent node pointer if y == itnil else if node.interval.low < y.interval.low ||node.interval.low == y.interval.low &&node.interval.high <= y.interval.high) else node.left = it.nil sets the left and right child nodes to the sentinel nodes, nil noderight = it.nil node.maxhigh = max(node.maxhigh, node.interval.high) it.InsertFixup(node) Adjust red-black tree balance} Adjust red-black tree balance (on insertion)func (it *intervaltree) insertfixup(z *node) else case 3: z is the father's left child zp.color = "black" z.p.p.color = "red" it.rightrotate(z.p.p) }else else z.p.color = "black" z.p.p.color = "red" it.leftrotate(z.p.p) }it.root.color = "black"The root node must be black} left-handed operation func (it *intervaltree) leftrotate(x *node) yp = x.p if x == it.root else if x == x.p.left else y.left=x x.p=y updates the maxhigh value xmaxhigh=max(x.interval.high,max(x.left.maxhigh,x.right.maxhigh)) y.maxhigh=max(y.interval.high,max(y. left .max high,y.r ight .max high))} Right-hand operation func (it *intervaltree) rightrotate(y *node) xp = y.p if y == it.root else if y == y.p.left else x. right=y y .p=x updates the maxhigh value ymaxhigh=max(y.interval.high,max(y.left.maxhigh,y.right.maxhigh)) x.maxhigh=max(x.interval.high,max(x.left.maxhigh,x.right.maxhigh)) interval-search-exactly: Find the exact same interval in t func (it *intervaltree) intervalsearchexactly(node *node, i interval) *node if ilow < node.interval.low ||i.low == node.interval.low &&i.high <= node.interval.high) return it.intervalsearchexactly(node.right, i)} Helper function: returns the greater of two integers func max(a, b int) int return b} test**func main() root: nil, }intervals := interval, ,for ,interval := range intervals test interval-search-exactly operation exactinterval := interval foundnode := it.intervalsearchexactly(it.root, exactinterval) if foundnode == it.nil else }Note that to simplify the example, I've omitted the handling of some boundary cases, such as node color attributes, the process of adjusting balance, and so on. You can modify and scale as needed to meet more complex needs.
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In go, in order to support the intervalsearch-exactly operation, we need to modify the search process of the interval tree. The goal of this operation is to find nodes that exactly match the given interval i. If no such node exists, we return tnil。Here's a possible implementation:
package mainimport ( "fmt"Type IntervalNode struct var tnil *IntervalNode sentinel node, which indicates a non-existent node func intervalsearch exactly(root *intervalnode, i int) *intervalnode If the interval of the current node exactly matches the given interval i, return the current node if rootintlow == i &&root.inthigh == i If the low endpoint of the current node is greater than i, then the matching interval can only be in the left subtree if rootintlow > i If the high end of the current node is less than i, then the matching interval can only be in the right subtree if rootinthigh < i If the interval of the current node contains i, then we need to continue searching in the left and right subtrees leftresult := intervalsearch exactly(root.left, i) rightresult := intervalsearch_exactly(root.Right, i) If no matching interval is found in the left and right subtrees, return tnil if leftresult == tnil &&&rightresult == tnil If there is only one result in the left and right subtrees, return that result if leftresult == tnil if rightresult == tnil If both left and right subtrees have results, this is not possible because the interval tree is bipartite Here we can add logic to handle this situation, e.g. return the result of the left subtree or the result of the right subtree Here we assume that the result of the left subtree is better, and return the result of the left subtree return leftresult}func main() to build the interval tree t .Given interval i i := 5 Perform intervalsearch-exactly operation result := intervalsearch exactly(troot, i) if result != tnil else }Note that this implementation assumes that the interval tree is balanced and that the low and high endpoints are unique for each node. In the actual go, you may need to adjust this according to the specific interval tree implementation.
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In order to support the new operation intervalsearch-exactly(t, i), we need to modify the data structure of the interval tree and the related operations. Here's the modified Go implementation:
package mainimport "fmt"type interval struct type node struct func newnode(interval interval) *node }func insert(root *node, interval interval) *node if interval.start< root.interval.start else return root}func intervalsearchexactly(root *node, i interval) *node if i.start< current.interval.start else }return nil}func main() root.left = newnode(interval) root.right = newnode(interval) i := interval result := intervalsearchexactly(root, i) if result != nil else }Note that this implementation assumes that the intervals in the interval tree are unique. If there may be duplicate intervals in the interval tree, you need to modify them accordingly according to your specific needs.
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