diff --git a/examples/IndependentSet.jl b/examples/IndependentSet.jl
index 65a5dcf9..3d488553 100644
--- a/examples/IndependentSet.jl
+++ b/examples/IndependentSet.jl
@@ -32,7 +32,7 @@ show_graph(graph; locs=locations)
 #     x_i^{w_i}
 # \end{matrix}\right),
 # ```
-# where ``W(x_i)_0=1`` is the first element associated with ``s_i=0`` and ``W(x_i)_1=x_i^{w_i}`` is the second element associated with ``s_i=1``, and `w_i` is the weight of vertex ``i``.
+# where ``W(x_i)_0=1`` is the first element associated with ``s_i=0`` and ``W(x_i)_1=x_i^{w_i}`` is the second element associated with ``s_i=1``, and ``w_i`` is the weight of vertex ``i``.
 # Similarly, on each edge ``(u, v)``, we define a matrix ``B`` indexed by ``s_u`` and ``s_v`` as
 # ```math
 # B = \left(\begin{matrix}
diff --git a/src/arithematics.jl b/src/arithematics.jl
index 35b7d25f..15924622 100644
--- a/src/arithematics.jl
+++ b/src/arithematics.jl
@@ -221,42 +221,106 @@ function Base.:*(a::ExtendedTropical{K,TO}, b::ExtendedTropical{K,TO}) where {K,
     return ExtendedTropical{K,TO}(sorted_sum_combination!(res, a.orders, b.orders))
 end
 
+# 1. bisect over summed value and find the critical value `c`,
+# 2. collect the values with sum combination `≥ c`,
+# 3. sort the collected values
 function sorted_sum_combination!(res::AbstractVector{TO}, A::AbstractVector{TO}, B::AbstractVector{TO}) where TO
     K = length(res)
     @assert length(B) == length(A) == K
-    @inbounds maxval = A[K] * B[K]
-    ptr = K
-    @inbounds res[ptr] = maxval
-    @inbounds queue = [(K,K-1,A[K]*B[K-1]), (K-1,K,A[K-1]*B[K])]
-    for k = 1:K-1
-        @inbounds (i, j, res[K-k]) = _pop_max_sum!(queue)   # TODO: do not enumerate, use better data structures
-        _push_if_not_exists!(queue, i, j-1, A, B)
-        _push_if_not_exists!(queue, i-1, j, A, B)
+    @inbounds high = A[K] * B[K]
+
+    mA = findfirst(!iszero, A)
+    mB = findfirst(!iszero, B)
+    if mA === nothing || mB === nothing
+        res .= Ref(zero(TO))
+        return res
+    end
+    @inbounds low = A[mA] * B[mB]
+    # count number bigger than x
+    c, _ = count_geq(A, B, mB, low, true)
+    @inbounds if c <= K   # return
+        res[K-c+1:K] .= sort!(collect_geq!(view(res,1:c), A, B, mB, low))
+        if c < K
+            res[1:K-c] .= zero(TO)
+        end
+        return res
+    end
+    # calculate by bisection for at most 30 times.
+    @inbounds for _ = 1:30
+        mid = mid_point(high, low)
+        c, nB = count_geq(A, B, mB, mid, true)
+        if c > K
+            low = mid
+            mB = nB
+        elseif c == K  # return
+            # NOTE: this is the bottleneck
+            return sort!(collect_geq!(res, A, B, mB, mid))
+        else
+            high = mid
+        end
     end
+    clow, _ = count_geq(A, B, mB, low, false)
+    @inbounds res .= sort!(collect_geq!(similar(res, clow), A, B, mB, low))[end-K+1:end]
     return res
 end
 
-function _push_if_not_exists!(queue, i, j, A, B)
-    @inbounds if j>=1 && i>=1 && !any(x->x[1] >= i && x[2] >= j, queue)
-        push!(queue, (i, j, A[i]*B[j]))
+# count the number of sum-combinations with the sum >= low
+function count_geq(A, B, mB, low, earlybreak)
+    K = length(A)
+    k = 1   # TODO: we should tighten mA, mB later!
+    @inbounds Ak = A[K-k+1]
+    @inbounds Bq = B[K-mB+1]
+    c = 0
+    nB = mB
+    @inbounds for q = K-mB+1:-1:1
+        Bq = B[K-q+1]
+        while k < K && Ak * Bq >= low
+            k += 1
+            Ak = A[K-k+1]
+        end
+        if Ak * Bq >= low
+            c += k
+        else
+            c += (k-1)
+            if k==1
+                nB += 1
+            end
+        end
+        if earlybreak && c > K
+            return c, nB
+        end
     end
+    return c, nB
 end
 
-function _pop_max_sum!(queue)
-    maxsum = first(queue)[3]
-    maxloc = 1
-    @inbounds for i=2:length(queue)
-        m = queue[i][3]
-        if m > maxsum
-            maxsum = m
-            maxloc = i
+function collect_geq!(res, A, B, mB, low)
+    K = length(A)
+    k = 1   # TODO: we should tighten mA, mB later!
+    Ak = A[K-k+1]
+    Bq = B[K-mB+1]
+    l = 0
+    for q = K-mB+1:-1:1
+        Bq = B[K-q+1]
+        while k < K && Ak * Bq >= low
+            k += 1
+            Ak = A[K-k+1]
+        end
+        # push data
+        ck = Ak * Bq >= low ? k : k-1
+        for j=1:ck
+            l += 1
+            res[l] = Bq * A[end-j+1]
         end
     end
-    @inbounds data = queue[maxloc]
-    deleteat!(queue, maxloc)
-    return data
+    return res
 end
 
+# for bisection
+mid_point(a::Tropical{T}, b::Tropical{T}) where T = Tropical{T}((a.n + b.n) / 2)
+mid_point(a::CountingTropical{T,CT}, b::CountingTropical{T,CT}) where {T,CT} = CountingTropical{T,CT}((a.n + b.n) / 2, a.c)
+mid_point(a::Tropical{T}, b::Tropical{T}) where T<:Integer = Tropical{T}((a.n + b.n) ÷ 2)
+mid_point(a::CountingTropical{T,CT}, b::CountingTropical{T,CT}) where {T<:Integer,CT} = CountingTropical{T,CT}((a.n + b.n) ÷ 2, a.c)
+
 function Base.:+(a::ExtendedTropical{K,TO}, b::ExtendedTropical{K,TO}) where {K,TO}
     res = Vector{TO}(undef, K)
     ptr1, ptr2 = K, K
diff --git a/src/bounding.jl b/src/bounding.jl
index 99426924..be66b553 100644
--- a/src/bounding.jl
+++ b/src/bounding.jl
@@ -60,7 +60,7 @@ struct CacheTree{T}
 end
 function cached_einsum(se::SlicedEinsum, @nospecialize(xs), size_dict)
     if length(se.slicing) != 0
-        @warn "Slicing is not supported for caching! Fallback to `NestedEinsum`."
+        @warn "Slicing is not supported for caching, got nslices = $(length(se.slicing))! Fallback to `NestedEinsum`."
     end
     return cached_einsum(se.eins, xs, size_dict)
 end
diff --git a/test/arithematics.jl b/test/arithematics.jl
index b7059fee..036f413f 100644
--- a/test/arithematics.jl
+++ b/test/arithematics.jl
@@ -153,6 +153,17 @@ end
     end
 end
 
+@testset "count geq" begin
+    A = collect(1:10)
+    B = collect(2:2:20)
+    low = 20
+    c, _ = GraphTensorNetworks.count_geq(A, B, 1, low, false)
+    @test c == count(x->x>=low, vec([a*b for a in A, b in B]))
+    res = similar(A, c)
+    @test sort!(GraphTensorNetworks.collect_geq!(res, A, B, 1, low)) == sort!(filter(x->x>=low, vec([a*b for a in A, b in B])))
+end
+
+
 # check the correctness of sampling
 @testset "generate samples" begin
     Random.seed!(2)