Added sqrt decomposition (#613)

* Create binary_heap.go

* Update binary_heap.go

* fix

* Update binary_heap.go

* Create sqrt_decomposition_simple.go

* move files

* bono

* bono

* fix style

* fix style

* .

* some changes

* .

* .

sqrt/sqrtdecomposition.go

@@ -0,0 +1,102 @@
+// Package sqrt contains algorithms and data structures that contains a √n in their complexity
+package sqrt
+
+import "math"
+
+// Sqrt (or Square Root) Decomposition is a technique used for query an array and perform updates
+// Inside this package is described its most simple data structure, you can find more at: https://cp-algorithms.com/data_structures/sqrt_decomposition.html
+//
+// Formally, You can use SqrtDecomposition only if:
+//
+// Given a function $Query:E_1,...,E_n\rightarrow Q$
+//
+// if $\exist unionQ:Q,Q\rightarrow Q$
+//
+// s.t.
+//
+// - $\forall n\in \N > 1, 1\le i<n, E_1,..., E_n\in E \\ query(E_1,..., E_n)=unionQ(query(E_1,..., E_i), query(E_{i+1},...,E_n))$
+//
+// - (Only if you want use $update$ function)
+// $\forall n\in \N > 0, E_1,..., E_n\in E \\ query(E_1,...,E_{new},..., E_n)=updateQ(query(E_1,...,E_{old},...,E_n), indexof(E_{old}), E_{new})$
+type SqrtDecomposition[E any, Q any] struct {
+	querySingleElement func(element E) Q
+	unionQ             func(q1 Q, q2 Q) Q
+	updateQ            func(oldQ Q, oldE E, newE E) (newQ Q)
+
+	elements  []E
+	blocks    []Q
+	blockSize uint64
+}
+
+// Create a new SqrtDecomposition instance with the parameters as specified by SqrtDecomposition comment
+// Assumptions:
+//   - len(elements) > 0
+func NewSqrtDecomposition[E any, Q any](
+	elements []E,
+	querySingleElement func(element E) Q,
+	unionQ func(q1 Q, q2 Q) Q,
+	updateQ func(oldQ Q, oldE E, newE E) (newQ Q),
+) *SqrtDecomposition[E, Q] {
+	sqrtDec := &SqrtDecomposition[E, Q]{
+		querySingleElement: querySingleElement,
+		unionQ:             unionQ,
+		updateQ:            updateQ,
+		elements:           elements,
+	}
+	sqrt := math.Sqrt(float64(len(sqrtDec.elements)))
+	blockSize := uint64(sqrt)
+	numBlocks := uint64(math.Ceil(float64(len(elements)) / float64(blockSize)))
+	sqrtDec.blocks = make([]Q, numBlocks)
+	for i := uint64(0); i < uint64(len(elements)); i++ {
+		if i%blockSize == 0 {
+			sqrtDec.blocks[i/blockSize] = sqrtDec.querySingleElement(elements[i])
+		} else {
+			sqrtDec.blocks[i/blockSize] = sqrtDec.unionQ(sqrtDec.blocks[i/blockSize], sqrtDec.querySingleElement(elements[i]))
+		}
+	}
+	sqrtDec.blockSize = blockSize
+	return sqrtDec
+}
+
+// Performs a query from index start to index end (non included)
+// Assumptions:
+//   - start < end
+//   - start and end are valid
+func (s *SqrtDecomposition[E, Q]) Query(start uint64, end uint64) Q {
+	firstIndexNextBlock := ((start / s.blockSize) + 1) * s.blockSize
+	q := s.querySingleElement(s.elements[start])
+	if firstIndexNextBlock > end { // if in same block
+		start++
+		for start < end {
+			q = s.unionQ(q, s.querySingleElement(s.elements[start]))
+			start++
+		}
+	} else {
+		// left side
+		start++
+		for start < firstIndexNextBlock {
+			q = s.unionQ(q, s.querySingleElement(s.elements[start]))
+			start++
+		}
+
+		//middle part
+		endBlock := end / s.blockSize
+		for i := firstIndexNextBlock / s.blockSize; i < endBlock; i++ {
+			q = s.unionQ(q, s.blocks[i])
+		}
+
+		// right part
+		for i := endBlock * s.blockSize; i < end; i++ {
+			q = s.unionQ(q, s.querySingleElement(s.elements[i]))
+		}
+	}
+	return q
+}
+
+// Assumptions:
+//   - index is valid
+func (s *SqrtDecomposition[E, Q]) Update(index uint64, newElement E) {
+	i := index / s.blockSize
+	s.blocks[i] = s.updateQ(s.blocks[i], s.elements[index], newElement)
+	s.elements[index] = newElement
+}

sqrt/sqrtdecomposition_test.go

@@ -0,0 +1,80 @@
+package sqrt_test
+
+import (
+	"github.com/TheAlgorithms/Go/sqrt"
+	"testing"
+)
+
+// Query interval
+type query struct {
+	firstIndex uint64
+	lastIndex  uint64
+}
+
+type update struct {
+	index uint64
+	value int
+}
+
+func TestSqrtDecomposition(t *testing.T) {
+	var sqrtDecompositionTestData = []struct {
+		description string
+		array       []int
+		updates     []update
+		queries     []query
+		expected    []int
+	}{
+		{
+			description: "test 1-sized array",
+			array:       []int{1},
+			queries:     []query{{0, 1}},
+			expected:    []int{1},
+		},
+		{
+			description: "test array with size 5",
+			array:       []int{1, 2, 3, 4, 5},
+			queries:     []query{{0, 5}, {0, 2}, {2, 4}},
+			expected:    []int{15, 3, 7},
+		},
+		{
+			description: "test array with size 5 and updates",
+			array:       []int{1, 2, 3, 4, 5},
+			updates: []update{{index: 1, value: 3},
+				{index: 2, value: 4}},
+			queries:  []query{{0, 5}, {0, 2}, {2, 4}},
+			expected: []int{17, 4, 8},
+		},
+		{
+			description: "test array with size 11 and updates",
+			array:       []int{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
+			updates: []update{{index: 2, value: 2},
+				{index: 3, value: 3},
+				{index: 6, value: 6}},
+			queries:  []query{{3, 5}, {7, 8}, {3, 7}, {0, 10}},
+			expected: []int{4, 1, 11, 18},
+		},
+	}
+	for _, test := range sqrtDecompositionTestData {
+		t.Run(test.description, func(t *testing.T) {
+			s := sqrt.NewSqrtDecomposition(test.array,
+				func(e int) int { return e },
+				func(q1, q2 int) int { return q1 + q2 },
+				func(q, a, b int) int { return q - a + b },
+			)
+
+			for i := 0; i < len(test.updates); i++ {
+				s.Update(test.updates[i].index, test.updates[i].value)
+			}
+
+			for i := 0; i < len(test.queries); i++ {
+				result := s.Query(test.queries[i].firstIndex, test.queries[i].lastIndex)
+
+				if result != test.expected[i] {
+					t.Logf("FAIL: %s", test.description)
+					t.Fatalf("Expected result: %d\nFound: %d\n", test.expected[i], result)
+				}
+			}
+
+		})
+	}
+}