diff --git a/imperative/python/src/tensor.h b/imperative/python/src/tensor.h
index d3f2c9fffa3f0e24e584f211f7403ca10b327abd..5860a5c3685a913cd826ebc3093ec869c82865f8 100644
--- a/imperative/python/src/tensor.h
+++ b/imperative/python/src/tensor.h
@@ -309,7 +309,7 @@ inline auto apply(Subgraph graph, Tensor*const* args, size_t nargs) {
     for (size_t i = 0; i < nargs; ++i) {
         inputs.push_back(args[i]->shared_from_this());
     }
-    auto apply_functor = [](std::shared_ptr<OpDef> op, SmallVector<std::shared_ptr<Tensor>> inputs) {
+    auto apply_functor = [](std::shared_ptr<OpDef> op, SmallVector<std::shared_ptr<Tensor>> inputs, size_t) {
         return apply(op, std::move(inputs));
     };
     return graph.apply(inputs, apply_functor, &make_const);
@@ -317,7 +317,7 @@ inline auto apply(Subgraph graph, Tensor*const* args, size_t nargs) {
 
 template <typename T>
 auto apply(Subgraph graph, T&& tensors)
-        -> std::enable_if_t<std::is_same_v<decltype(tensors[0]), Tensor*>,
+        -> std::enable_if_t<std::is_same_v<std::decay_t<decltype(tensors[0])>, Tensor*>,
                             apply_result_t> {
     size_t nargs = tensors.size();
     Tensor* args[nargs];
diff --git a/imperative/src/impl/subgraph.cpp b/imperative/src/impl/subgraph.cpp
new file mode 100644
index 0000000000000000000000000000000000000000..087bd1fbcb0b3f2e10ab2aa41d77cf066738a077
--- /dev/null
+++ b/imperative/src/impl/subgraph.cpp
@@ -0,0 +1,105 @@
+/**
+ * \file imperative/src/impl/subgraph.cpp
+ * MegEngine is Licensed under the Apache License, Version 2.0 (the "License")
+ *
+ * Copyright (c) 2014-2021 Megvii Inc. All rights reserved.
+ *
+ * Unless required by applicable law or agreed to in writing,
+ * software distributed under the License is distributed on an
+ * "AS IS" BASIS, WITHOUT ARRANTIES OR CONDITIONS OF ANY KIND, either express or
+ * implied.
+ */
+
+#include "megbrain/imperative/subgraph.h"
+
+namespace mgb {
+namespace imperative {
+
+void Subgraph::remove_unused_exprs() {
+    std::unordered_set<size_t> required_vars = {outputs.begin(), outputs.end()};
+    required_vars.erase(0);
+    for (auto iter = exprs.rbegin(); iter != exprs.rend(); ++iter) {
+        auto& expr = *iter;
+        bool required = false;
+        for (auto output : expr.outputs) {
+            if (required_vars.count(output)) {
+                required = true;
+                break;
+            }
+        }
+        if (required) {
+            required_vars.insert(expr.inputs.begin(), expr.inputs.end());
+        } else {
+            expr.op = nullptr;
+        }
+    }
+    exprs.erase(std::remove_if(exprs.begin(), exprs.end(),
+                               [](auto expr) { return expr.op == nullptr; }),
+                exprs.end());
+}
+
+SmallVector<bool> Subgraph::gen_input_mask() {
+    std::unordered_set<size_t> unused_inputs = {inputs.begin(), inputs.end()};
+    for (auto&& expr : exprs) {
+        for (auto&& input : expr.inputs) {
+            unused_inputs.erase(input);
+        }
+    }
+    for (auto&& output : outputs) {
+        unused_inputs.erase(output);
+    }
+    unused_inputs.insert(0);
+    SmallVector<bool> mask(inputs.size(), true);
+    for (size_t i = 0; i < inputs.size(); ++i) {
+        if (unused_inputs.count(inputs[i])) {
+            mask[i] = false;
+        }
+    }
+    return mask;
+}
+
+SmallVector<bool> Subgraph::gen_output_mask() {
+    std::unordered_set<size_t> invalid_outputs = {outputs.begin(),
+                                                  outputs.end()};
+    for (auto&& input : inputs) {
+        invalid_outputs.erase(input);
+    }
+    for (auto&& expr : exprs) {
+        for (auto&& output : expr.outputs) {
+            invalid_outputs.erase(output);
+        }
+    }
+    for (auto&& constant: constants) {
+        invalid_outputs.erase(constant.first);
+    }
+    invalid_outputs.insert(0);
+    SmallVector<bool> mask(outputs.size(), true);
+    for (size_t i = 0; i < outputs.size(); ++i) {
+        if (invalid_outputs.count(outputs[i])) {
+            mask[i] = false;
+        }
+    }
+    return mask;
+}
+
+void Subgraph::replace_vars(
+        const std::unordered_map<size_t, size_t>& replace_map) {
+    // FIXME: preprocess replace_map
+    auto replace_var = [&](var_t& var) {
+        // TODO: detect infinite loop
+        while (replace_map.count(var)) {
+            var = replace_map.at(var);
+        }
+    };
+    for (auto& expr : exprs) {
+        for (auto& input : expr.inputs) {
+            replace_var(input);
+        }
+    }
+    for (auto& output : outputs) {
+        replace_var(output);
+    }
+}
+
+}  // namespace imperative
+}  // namespace mgb
diff --git a/imperative/src/include/megbrain/imperative/op_def.h b/imperative/src/include/megbrain/imperative/op_def.h
index 8b8641a51b66a3f189b12d4b517ed058054883f4..7ab725070f777b84f2759a298d7e6de396f9e0c5 100644
--- a/imperative/src/include/megbrain/imperative/op_def.h
+++ b/imperative/src/include/megbrain/imperative/op_def.h
@@ -14,6 +14,7 @@
 #include "megbrain/graph.h"
 #include "megbrain/imperative/physical_tensor.h"
 #include "megbrain/imperative/utils/to_string.h"
+#include "megbrain/imperative/subgraph.h"
 
 namespace mgb {
 namespace imperative {
@@ -28,54 +29,6 @@ enum DispatchMode {
 
 using SharedOp = std::shared_ptr<OpDef>;
 
-template <typename T>
-struct Expr {
-    std::shared_ptr<OpDef> op;
-    SmallVector<T> inputs;
-    SmallVector<T> outputs;
-};
-
-struct Subgraph {
-    SmallVector<size_t> inputs;
-    SmallVector<std::pair<size_t, TensorPtr>> constants;
-    SmallVector<size_t> outputs;
-    SmallVector<Expr<size_t>> exprs;
-
-    template <typename T, typename F, typename C>
-    SmallVector<T> apply(SmallVector<T> input_vars, F&& f, C&& c) const {
-        std::unordered_map<size_t, T> idx2var;
-        mgb_assert(inputs.size() == input_vars.size(), "input size mismatch");
-        for (size_t i = 0; i < inputs.size(); ++i) {
-            idx2var[inputs[i]] = input_vars[i];
-        }
-        for (auto&& [idx, val]: constants) {
-            idx2var[idx] = c(val);
-        }
-        for (auto& expr: exprs) {
-            SmallVector<T> expr_inputs;
-            for (auto idx: expr.inputs) {
-                expr_inputs.push_back(idx2var[idx]);
-            }
-            SmallVector<T> expr_outputs = f(expr.op, std::move(expr_inputs));
-            mgb_assert(expr_outputs.size() == expr.outputs.size(), "output size mismatch");
-            for (size_t i = 0; i < expr_outputs.size(); ++i) {
-                idx2var[expr.outputs[i]] = expr_outputs[i];
-            }
-        }
-        SmallVector<T> output_vars;
-        for (auto idx: outputs) {
-            output_vars.push_back(idx2var[idx]);
-        }
-        return output_vars;
-    }
-
-    bool empty() const {
-        return outputs.size() == 0;
-    }
-
-    std::string repr() const;
-};
-
 struct BackwardGraphResult {
     Subgraph backward;
     SmallVector<bool> save_for_backward;
diff --git a/imperative/src/include/megbrain/imperative/subgraph.h b/imperative/src/include/megbrain/imperative/subgraph.h
new file mode 100644
index 0000000000000000000000000000000000000000..68221e4538d1db09d5c32430ce6a1629fd66bae1
--- /dev/null
+++ b/imperative/src/include/megbrain/imperative/subgraph.h
@@ -0,0 +1,100 @@
+/**
+ * \file imperative/src/include/megbrain/imperative/subgraph.h
+ * MegEngine is Licensed under the Apache License, Version 2.0 (the "License")
+ *
+ * Copyright (c) 2014-2021 Megvii Inc. All rights reserved.
+ *
+ * Unless required by applicable law or agreed to in writing,
+ * software distributed under the License is distributed on an
+ * "AS IS" BASIS, WITHOUT ARRANTIES OR CONDITIONS OF ANY KIND, either express or
+ * implied.
+ */
+
+#pragma once
+
+#include <list>
+
+#include "megbrain/imperative/physical_tensor.h"
+#include "megbrain/imperative/utils/to_string.h"
+#include "megbrain/utils/small_vector.h"
+
+namespace mgb {
+namespace imperative {
+
+class OpDef;
+
+template <typename T>
+struct Expr {
+    std::shared_ptr<OpDef> op;
+    SmallVector<T> inputs;
+    SmallVector<T> outputs;
+};
+
+template <typename T>
+struct ToStringTrait<Expr<T>> {
+    std::string operator()(const Expr<T>& expr) {
+        return ssprintf("%s = %s %s\n", to_string(expr.inputs).c_str(), to_string(expr.op.get()).c_str(), to_string(expr.outputs).c_str());
+    }
+};
+
+struct Subgraph {
+    
+    template <typename TDesc>
+    class Builder;
+
+    using var_t = size_t;
+    using vars_t = SmallVector<size_t>;
+    using op_t = std::shared_ptr<OpDef>;
+    using expr_t = Expr<var_t>;
+
+    template <typename TDesc>
+    using builder_t = Builder<TDesc>;
+    
+    SmallVector<var_t> inputs;
+    SmallVector<std::pair<var_t, TensorPtr>> constants;
+    SmallVector<var_t> outputs;
+    SmallVector<expr_t> exprs;
+
+    template <typename T, typename F, typename C>
+    SmallVector<T> apply(SmallVector<T> input_vars, F&& f, C&& c) const {
+        std::unordered_map<size_t, T> idx2var;
+        mgb_assert(inputs.size() == input_vars.size(), "input size mismatch");
+        for (size_t i = 0; i < inputs.size(); ++i) {
+            idx2var[inputs[i]] = input_vars[i];
+        }
+        for (auto&& [idx, val] : constants) {
+            idx2var[idx] = c(val);
+        }
+        for (auto& expr : exprs) {
+            SmallVector<T> expr_inputs;
+            for (auto idx : expr.inputs) {
+                expr_inputs.push_back(idx2var[idx]);
+            }
+            SmallVector<T> expr_outputs =
+                    f(expr.op, std::move(expr_inputs), expr.outputs.size());
+            mgb_assert(expr_outputs.size() == expr.outputs.size(),
+                       "output size mismatch");
+            for (size_t i = 0; i < expr_outputs.size(); ++i) {
+                idx2var[expr.outputs[i]] = expr_outputs[i];
+            }
+        }
+        SmallVector<T> output_vars;
+        for (auto idx : outputs) {
+            output_vars.push_back(idx2var[idx]);
+        }
+        return output_vars;
+    }
+
+    void remove_unused_exprs();
+    SmallVector<bool> gen_input_mask();
+    SmallVector<bool> gen_output_mask();
+    bool empty() const { return outputs.size() == 0; }
+    void replace_vars(const std::unordered_map<size_t, size_t>& replace_map);
+    std::string repr() const;
+    bool is_single() const;
+    std::shared_ptr<OpDef> as_single() const;
+    bool operator==(const Subgraph& rhs) const;
+};
+
+}  // namespace imperative
+}  // namespace mgb
\ No newline at end of file