qbqp

.gitignore

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+/target

Cargo.lock

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+# This file is automatically @generated by Cargo.
+# It is not intended for manual editing.
+version = 3
+
+[[package]]
+name = "pbqp"
+version = "0.1.0"

Cargo.toml

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+[package]
+name = "pbqp"
+version = "0.1.0"
+edition = "2021"
+
+# See more keys and their definitions at https://doc.rust-lang.org/cargo/reference/manifest.html
+
+[dependencies]

README.md

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+# QBQP 算法
+
+寄存器分配算法。
+
+## 约束方程的含义
+
+主要就是约束方程了：
+
+$$
+min\{ \sum_{i \le i \lt j \le n} X_i^T C_{ij} X_j + \sum_{i=1}^n s_i X_i \}
+$$
+
+- $X_i = (x_{i, sp}, x_{i, r_1}, x_{i, r_2}, ..., x_{i, r_K})^T$
+  - 这里 $x_{i, sp}$ 表示将 $t_i$ spill -> stack 的代价。
+  - $X_i = (x, xxx, ..., x)^T$ 只有一个 分量=1，其他都是 0
+  - K 表示 $t_i$ 有 K 个可选的寄存器(不一定是所有的 physical reg，这是显然的，比方说不想要破坏到 callee-saved 寄存器)。
+- $C_{ij}$ 量化了 $t_i$ 和 $t_j$ 之间的相互影响
+  - i === j? (不一定，比方说在传参的时候，i 可以放到 a0， 但是 j 是不能放到 a0 的)
+  - 但是我有一种想法: 就是我认为还是要把 32 个寄存器全部摆上台面的，如果不能用，我认为可以相应的可以设置为 $\infin$ ，这显然可能会变成一个 稀疏矩阵，然后是不是可以进行 凸优化啥的。如果 i!=j 并且有: $R_i$ ^ $R_j \ne \phi$ 那这可能需要对齐，可能有点麻烦
+  - 如果 i === j，那么可以做一件事情：将 $c_{ij}$ 的对角线设置为 负数。对角线意味着: virtual reg $t_i$ 和 $t_j$ 是可接合的，可以分配到 同一个 physical reg 的，负数的 语义 是：鼓励接合(代价总是越小越好)
+- $s_i = (\vec{cost(i)}, \vec0, ..., \vec0)$
+  - $\vec{cost(t_i)}$ : $t_i$(virtual reg) 溢出的代价。如果每个 $t_i$ 都选择了 spill ，那么相当于是: $\forall t_i, X_i = (1, 0, ..., 0)$，那么 $\sum_{i=1}^n s_i X_i = \sum_{i=1}^n cost(t_i)$
+
+## 伪代码
+
+伪代码 ( from 华保健 ) 如下：
+
+```pseudocode
+void reduce_1(vertex x) {
+	y = adj(x);
+	for (i = 1; i < len(E_y); i++) {
+		delta(i) = min(C_xy(i, :) + E_x);
+	}
+	E_y += delta;
+	remote_vertex(x);
+}
+
+void reduce_2(vertex x) {
+	y, z = adj(x);
+	for (i = 1; i < len(E_y); i++) {
+		for (j = 1; j < len(E_z); j++) {
+			delta(i) = min(C_xy(i, :) + C_xz(j, :) + E_x);
+		}
+	}
+	C_yz += delta;
+	remote_vertex(x);
+}
+
+void reduce_N(vertex x) {
+	y, z = adj(x);
+	for (i = 1; i < len(E_y); i++) {
+		delta(i) = 0;
+		for (j = 1; j < len(E_z); j++) {
+			delta(i) = min(C_xy(i, :) + E_y);
+		}
+	}
+	min_index = get_min_index(delta);
+	for y : vertex in adj(x) {
+		E_y += C_xy(min_index, :);
+	}
+	remote_vertex(x);
+}
+
+void solve_PBQP(graph pbqp_model) {
+	while (pbqp_model still has edges) {
+		simplify(pbqp_model);
+		while (there exists any vertex x of degree 1) {
+			reduce_1(x);
+		}
+		while (there exists any vertex x of degree 2) {
+			reduce_2(x);
+		}
+		while (there exists any vertex x of degree >= 2) {
+			reduce_N(x);
+		}
+	}
+}
+```
+
+### rust
+
+看 src/main.rs (from gpt)

src/main.rs

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+use std::collections::{HashMap, HashSet};
+
+struct Vertex {
+    cost_vector: Vec<i32>,     // 代价向量
+    neighbors: HashSet<usize>, // 邻接点索引集
+}
+
+struct Edge {
+    cost_matrix: Vec<Vec<i32>>, // 代价矩阵
+}
+
+struct Graph {
+    vertices: Vec<Vertex>,
+    edges: HashMap<(usize, usize), Edge>,
+}
+
+impl Graph {
+    fn new() -> Self {
+        Self {
+            vertices: vec![],
+            edges: HashMap::new(),
+        }
+    }
+
+    fn add_vertex(&mut self, cost_vector: Vec<i32>) -> usize {
+        let index = self.vertices.len();
+        self.vertices.push(Vertex {
+            cost_vector,
+            neighbors: HashSet::new(),
+        });
+        index
+    }
+
+    fn add_edge(&mut self, v1: usize, v2: usize, cost_matrix: Vec<Vec<i32>>) {
+        self.vertices[v1].neighbors.insert(v2);
+        self.vertices[v2].neighbors.insert(v1);
+        self.edges.insert(
+            (v1, v2),
+            Edge {
+                cost_matrix: cost_matrix.clone(),
+            },
+        );
+        self.edges.insert((v2, v1), Edge { cost_matrix });
+    }
+
+    fn remove_vertex(&mut self, index: usize) {
+        if let Some(vertex) = self.vertices.get(index) {
+            for &neighbor in &vertex.neighbors {
+                self.vertices[neighbor].neighbors.remove(&index);
+                self.edges.remove(&(index, neighbor));
+                self.edges.remove(&(neighbor, index));
+            }
+        }
+        self.vertices.remove(index);
+    }
+
+    // Function to simplify the graph based on degree of vertices
+    fn simplify(&mut self) {
+        while !self.edges.is_empty() {
+            for i in 0..self.vertices.len() {
+                match self.vertices[i].neighbors.len() {
+                    1 => self.reduce_1(i),
+                    2 => self.reduce_2(i),
+                    _ => self.reduce_n(i),
+                }
+            }
+        }
+    }
+
+    fn reduce_1(&mut self, x: usize) {
+        let y = *self.vertices[x].neighbors.iter().next().unwrap();
+        let edge = self.edges.get(&(x, y)).unwrap();
+        let mut delta = Vec::new();
+
+        for i in 0..self.vertices[y].cost_vector.len() {
+            let min_cost = edge.cost_matrix[i].iter().min().unwrap();
+            delta.push(min_cost + self.vertices[x].cost_vector[i]);
+        }
+
+        for i in 0..self.vertices[y].cost_vector.len() {
+            self.vertices[y].cost_vector[i] += delta[i];
+        }
+
+        self.remove_vertex(x);
+    }
+
+    fn reduce_2(&mut self, x: usize) {
+        let mut iter = self.vertices[x].neighbors.iter();
+        let y = *iter.next().unwrap();
+        let z = *iter.next().unwrap();
+        let mut delta = Vec::new();
+
+        for i in 0..self.vertices[y].cost_vector.len() {
+            for j in 0..self.vertices[z].cost_vector.len() {
+                let cost_y = self.edges.get(&(x, y)).unwrap().cost_matrix[i][j];
+                let cost_z = self.edges.get(&(x, z)).unwrap().cost_matrix[i][j];
+                delta.push(std::cmp::min(cost_y, cost_z) + self.vertices[x].cost_vector[i]);
+            }
+        }
+
+        for i in 0..self.vertices[y].cost_vector.len() {
+            self.vertices[y].cost_vector[i] += delta[i];
+            self.vertices[z].cost_vector[i] += delta[i];
+        }
+
+        self.remove_vertex(x);
+    }
+
+    fn reduce_n(&mut self, x: usize) {
+        // This function needs to be implemented based on specific problem constraints and conditions
+    }
+}
+
+fn main() {
+    let mut graph = Graph::new();
+    // Example usage
+    let v0 = graph.add_vertex(vec![1, 2, 3]);
+    let v1 = graph.add_vertex(vec![4, 5, 6]);
+    let v2 = graph.add_vertex(vec![7, 8, 9]);
+
+    graph.add_edge(v0, v1, vec![vec![10, 20], vec![30, 40]]);
+    graph.add_edge(v1, v2, vec![vec![50, 60], vec![70, 80]]);
+
+    graph.simplify();
+}