optimize Chain.Ops() (#99)

Use linear 2SUM-like algorithm in the typical case where the chain is
ascending.

Updates #60
Updates #25

chain.go

@@ -15,6 +15,9 @@ import (
 //	[efficientcompaddchain]  Bergeron, F., Berstel, J. and Brlek, S. Efficient computation of addition
 //	                         chains. Journal de theorie des nombres de Bordeaux. 1994.
 //	                         http://www.numdam.org/item/JTNB_1994__6_1_21_0
+//	[knuth]                  Knuth, Donald E. Evaluation of Powers. The Art of Computer Programming, Volume 2
+//	                         (Third Edition): Seminumerical Algorithms, chapter 4.6.3. 1997.
+//	                         https://www-cs-faculty.stanford.edu/~knuth/taocp.html
 
 // Chain is an addition chain.
 type Chain []*big.Int
@@ -49,6 +52,25 @@ func (c Chain) End() *big.Int {
 func (c Chain) Ops(k int) []Op {
 	ops := []Op{}
 	s := new(big.Int)
+
+	// If the prefix is ascending this can be done in linear time.
+	if c[:k].IsAscending() {
+		for l, r := 0, k-1; l <= r; {
+			s.Add(c[l], c[r])
+			cmp := s.Cmp(c[k])
+			if cmp == 0 {
+				ops = append(ops, Op{l, r})
+			}
+			if cmp <= 0 {
+				l++
+			} else {
+				r--
+			}
+		}
+		return ops
+	}
+
+	// Fallback to quadratic.
 	for i := 0; i < k; i++ {
 		for j := i; j < k; j++ {
 			s.Add(c[i], c[j])
@@ -57,6 +79,7 @@ func (c Chain) Ops(k int) []Op {
 			}
 		}
 	}
+
 	return ops
 }
 
@@ -135,6 +158,21 @@ func (c Chain) Superset(targets []*big.Int) error {
 	return nil
 }
 
+// IsAscending reports whether the chain is ascending, that is if it's in sorted
+// order without repeats, as defined in [knuth] Section 4.6.3 formula (11).
+// Does not fully validate the chain, only that it is ascending.
+func (c Chain) IsAscending() bool {
+	if len(c) == 0 || !bigint.EqualInt64(c[0], 1) {
+		return false
+	}
+	for i := 1; i < len(c); i++ {
+		if c[i-1].Cmp(c[i]) >= 0 {
+			return false
+		}
+	}
+	return true
+}
+
 // Product computes the product of two addition chains. The is the "o times"
 // operator defined in [efficientcompaddchain] Section 2.
 func Product(a, b Chain) Chain {

chain_test.go

@@ -5,6 +5,79 @@ import (
 	"testing"
 )
 
+func TestChainOps(t *testing.T) {
+	cases := []struct {
+		Name   string
+		Chain  Chain
+		Expect [][]Op
+	}{
+		{
+			Name:  "short",
+			Chain: Int64s(1, 2),
+			Expect: [][]Op{
+				{{0, 0}}, // 2
+			},
+		},
+		{
+			Name:  "multiple_choices",
+			Chain: Int64s(1, 2, 3, 4),
+			Expect: [][]Op{
+				{{0, 0}},         // 2
+				{{0, 1}},         // 3
+				{{0, 2}, {1, 1}}, // 4
+			},
+		},
+		{
+			Name:  "non_ascending",
+			Chain: Int64s(1, 2, 3, 4, 7, 5, 6),
+			Expect: [][]Op{
+				{{0, 0}},                 // 2
+				{{0, 1}},                 // 3
+				{{0, 2}, {1, 1}},         // 4
+				{{2, 3}},                 // 7
+				{{0, 3}, {1, 2}},         // 5
+				{{0, 5}, {1, 3}, {2, 2}}, // 6
+			},
+		},
+	}
+	for _, c := range cases {
+		c := c // scopelint
+		t.Run(c.Name, func(t *testing.T) {
+			var got [][]Op
+			for k := 1; k < len(c.Chain); k++ {
+				got = append(got, c.Chain.Ops(k))
+			}
+			if !reflect.DeepEqual(got, c.Expect) {
+				t.Logf("got    = %v", got)
+				t.Logf("expect = %v", c.Expect)
+				t.Fail()
+			}
+		})
+	}
+}
+
+func TestChainIsAscending(t *testing.T) {
+	cases := []struct {
+		Name   string
+		Chain  Chain
+		Expect bool
+	}{
+		{Name: "empty", Chain: Int64s(), Expect: false},
+		{Name: "does_not_start_with_one", Chain: Int64s(42), Expect: false},
+		{Name: "ascending", Chain: Int64s(1, 2, 3, 5, 8), Expect: true},
+		{Name: "repeat", Chain: Int64s(1, 2, 3, 3, 8), Expect: false},
+		{Name: "not_sorted", Chain: Int64s(1, 2, 3, 4, 7, 5, 6), Expect: false},
+	}
+	for _, c := range cases {
+		c := c // scopelint
+		t.Run(c.Name, func(t *testing.T) {
+			if got := c.Chain.IsAscending(); got != c.Expect {
+				t.Fatalf("%v.IsAscending() = %v; expect %v", c.Chain, got, c.Expect)
+			}
+		})
+	}
+}
+
 func TestProduct(t *testing.T) {
 	a := Int64s(1, 2, 4, 6, 10)
 	b := Int64s(1, 2, 4, 8)